1

CHAPTER 1

INTRODUCTION

1.1 FLUID DYNAMICS

Fluid dynamics is the science in which we study the properties of fluids (liquids and

gases) in motion. The systematic study of fluid dynamics or hydrodynamics started only after the

Euler’s discovery of the equations of motion of an inviscid fluid. An attempt to describe the

affect of fluid motion is made by Newton, who conceived the idea that the fluid consisted of a

granulated structure of discrete particles. Later, some significant contributions to this subject

were made by the following scientists. Langrange gave the concept of velocity potential stream

function. The principle of resistance to flow in capillary tubes was given by Poiseuille. The credit

for the equations of motion of viscous fluids goes to Navier and Stokes. Reynolds discovered the

equations of turbulence motion. Prandtl put forward the boundary layer theory. The theories of

turbulence and stabilities are the creations of G. I. Taylor and Lord Rayleigh. Still later, some

other excellent contributions were given by many more famous scientists/mathematicians which

include Bénard, Kutta, Prandtl, Lord Kelvin, Orr, Sommerfield, Rayleigh, Zhukovski and

Kármán etc. Now-a-days fluid dynamics has become a very vast subject and has given birth to

many other subjects like meteorology, gas dynamics, aerodynamics, non-Newtonian flows,

magnetohydrodynamics etc.

Fluid dynamics and its subdisciplines like aerodynamics, hydrodynamics and hydraulics

have a wide range of applications. Examples include the design of aircraft, calculating forces and

moments on aircraft, determining the mass flow rate of petroleum through pipelines, predicting

weather patterns and control of many industrial processes. Fluid dynamics is the key of our

understanding some of the most important phenomena in our physical world like ocean currents,

weather systems, convection currents such as motions of molten rocks inside the earth and the

motion in the outer layer of the sun and the swirling of gases in galaxies.

Classical (or Newtonian) mechanics and continuum hypothesis are going to act as the

basis for study of fluid dynamics. Classical mechanics uses the concept of point particles, objects

with negligible size. The motion of a point particle is characterized by a small number of

2

parameters: its position, mass and the forces applied to it. Classical mechanics is often referred to

as Newtonian mechanics after Newton and his laws of motion. We confine ourselves to

Newtonian mechanics and shall not evoke the theory of relativity. In other words, we restrict

ourselves to those systems where particle velocities are small in comparison to the velocity of

light so as to have negligible relativity effects. In this way, we are not concerned with those

masses, velocities and temperatures for which Newtonian mechanics does not provide adequate

description.

In classical fluid dynamics, the fluid molecules are considered electrically neutral. The

study of water flowing in rivers, waves in ocean and the motion of aero plane in the lower parts

of Earth’s atmosphere are in the domain of classical fluid dynamics. The gross properties of

various states of matter are directly related to the molecular structure and the nature of

intermolecular forces that operate between the constituent molecules. In solids, the arrangement

of molecules is virtually permanent and under normal conditions may have a simple periodic

structure as in case of crystals and are acted upon by strong intermolecular forces. The

arrangement of molecules in liquids is partially ordered and is acted upon by medium

intermolecular forces. In case of gases and plasmas, weak short-range intermolecular forces act

upon the particles and molecular arrangements are disordered.

1.2 CONTINUUM HYPOTHESIS

In fluid dynamics, we make use of continuum theory though we know that matter is

composed of atoms and molecules and therefore has necessarily a discrete structure. In normal

gases, the masses are concentrated in molecules. These molecules are separated by vacuous

regions with linear dimensions much larger than those of molecules themselves. In liquids and

solids, though the average spacing between the molecules and atoms is small, the masses are

concentrated in the nuclei of the atoms composing a molecule and are very far from being

smeared uniformly over the volume occupied by the liquid. When the fluid is viewed on

microscopic scale so as to reveal the individual molecules, the properties of fluid such as

composition, velocity and density have violently non-uniform distributions. Since we are

generally concerned with the macroscopic behaviour at the mass centres are smeared out

uniformly over a certain volume surrounding them and treat the matter as continuum. This is

called “continuum hypothesis”. There is ample evidence that common real fluids, both liquids

3

and gases move as if they were continuous under normal conditions and even under considerable

departure from normal conditions. The hypothesis is justified when we consider only those

systems in which the characteristic length is much larger than the mean free path of the fluid

molecules. The continuum approach is simpler than the more rigorous kinematic one. Because

our hypothesis has made it possible to give meaning to terms such as density, pressure,

temperature, momentum and angular momentum ‘at a point’. And, in general, the values of these

quantities are continuous functions of position and time, thus permitting us the use of derivatives

and differentials whenever they are needed.

The foundational axioms of fluid dynamics are the laws of conservation of mass,

conservation of momentum (also known as Newton’s second law or the balance law) and

conservation of energy. These are based on Classical mechanics and modified in Relativistic

mechanics. The central equations for fluid dynamics are the Navier-Stokes equations which are

non-linear differential equations describing the flow of a fluid whose stress depends linearly on

velocity and pressure.

The knowledge of thermo-hydrodynamics, mass transfer, heat transfer and

electromagnetic theory is being dealt in detail in fluid dynamics. In view of this, it is an important

subject for the investigators in engineering science (Yuan [1]). The heat transfer in fluid medium

can take place in three modes, namely conduction, convection and radiation. The thermal

convection in fluid can be classified as forced convection and free convection. Prior to recent

years the engineering applications of fluid mechanics were restricted to the systems in which the

electric and magnetic fields play no role. However, the interaction of electromagnetic fields and

fluids has been quite interesting in view of their large applications in fields like controlled nuclear

fusion, engineering, medicine and high speed silent printing etc. The study of various field and

fluid interactions may be divided into three main categories:

i. Electrohydrodynamics (EHD), the branch of fluid mechanics concerned with electric

force effects;

ii. Magnetohydrodynamics (MHD), the study of interaction between magnetic fields and

fluid conductors of electricity; and

iii. Ferrohydrodynamics (FHD), the study of interaction of magnetic fields and non-

conducting ferromagnetic fluids.

4

1.3 MAGNETOHYDRODYNAMICS (MHD)

Magnetohydrodynamics is the academic discipline which deals with the dynamics of

electrically conducting fluids. It is concerned with the motion of fluids that are good conductors

of electricity and specifically with those effects that arise through the interaction of the motion of

the fluid and any ambient magnetic field that may be present. Such a field is produced by electric

current sources which may be either external to the fluid or induced within the fluid itself. The set

of equations which describe MHD are a combination of the Navier-Stokes equations of fluid

dynamics and Maxwell’s equations of electromagnetism.

MHD is concerned with the physical systems specified by the equations that result from

the fusion of those of hydrodynamics and electromagnetic theory. It is well known fact that when

a conductor moves in a magnetic field, electric currents are induced init. These currents

experience a mechanical force, called Lorentz force, due to the presence of magnetic field. This

force tends to modify the initial motion of the conductor. Moreover, a magnetic field which is

generated by the induced currents is added to the applied magnetic field. Thus there is a coupling

between the motion of the conductor and electromagnetic field, which is exhibited in more

pronounced form in liquid and gaseous conductors. This is due to the fact that molecules

composing the liquids and gases enjoy more freedom of movement than those of solid

conductors. The Lorentz force is usually small unless inordinately high magnetic fields are

applied. Therefore this force is too small to alter the motion as a whole considerably but if it acts

for a sufficiently long period, the molecules of gases and liquids may get accelerated

considerably to change the initial state of motion of these types of conductors.

Magnetohydrodynamics is interesting from several standpoints. Ordinary fluids are

interesting and beautiful on their own, but magnetofluids have an extra property. Magnetofluids

can carry current which means that they can both generate field and can be influenced by

magnetic fields. This natural self interaction between the current and the magnetic field produces

some curious phenomena e.g. the behaviour of the solar magnetic field or the Earth’s magnetic

field.

A systematic study of magnetohydrodynamics was started by Alfvén [2]. Alfvén also

discovered the interlocking between mechanical forces and magnetic forces in a highly

conducting fluid moving in an external magnetic field and showed that this interaction would

5

produce a new kind of wave, which he named a MHD wave. It was pointed out by Batchelor [3]

that magnetic field imparts to the fluid a certain rigidity along with certain properties of elasticity

which enables it to transmit disturbances by new modes of wave propagation. Attempts to show

the existence of MHD waves in the laboratory were made by Lundquist [4] and Lehnert [5].

Progress in hydromagnetics, as in all physical sciences, depends upon the successful interaction

of theory and experiment.

The gradual development of magnetohydrodynamics has been exhibited in the work of

Sutton and Sherman [6], Roberts [7], Cowling [8], Bateman [9], Moffatt [10] and Chandrasekhar

[11].

MHD is related to engineering problems such as plasma confinement, liquid-metal

cooling of nuclear reactors and electromagnetic casting. Electromagnetic interactions of fluids

and plasmas are especially important to physicists in the study of stellar fusion and the solar

wind. It also finds some applications in the area of geophysics and astronomy.

1.4 FERROHYDRODYNAMICS (FHD)

Ferrohydrodynamics (FHD) deals with the mechanics of fluid motion influenced by

strong forces of magnetic polarization. In MHD the body force acting on the fluid is the Lorentz

force that arises when electric current flows at an angle to the direction of an impressed magnetic

field. However, in FHD usually no electric current is flowing in the fluid. The body force in FHD

is due to polarization force, which in turn requires material magnetization in the presence of

magnetic field gradients. In general, strong thermo mechanical coupling exists when the induced

polarization is both temperature and field dependent.

The importance of ferrohydrodynamics was realized soon after the method of formation

of ferrofluids. Ferrofluids do not exist in nature and are artificially prepared. In recent years,

researchers have prepared ferrofluids, which have the fluid properties of a liquid and the

magnetic properties of a solid. A ferrofluid is a suspension of fine magnetic particles (about 10

nm in diameter) in a liquid carrier (such as water or oil). Ferrohydrodynamics is of great interest

because the fluids of concern possess a giant magnetic response. The very well written

monograph by Rosenweig [12] is a perfect introduction to this fascinating subject. Rosenweig’s

book leads the reader through all areas of a research field i.e. from the synthesis of magnetic

fluids, their properties and the foundation of the theory of ferrohydrodynamics towards problems

6

of experimental hydrodynamics in ferrofluids. This monograph also reviews several applications

of heat transfer through ferrofluids. He very briefly refers to thermo-convective instability in

FHD.

The convective instability of a ferromagnetic fluid for a fluid layer heated from below in

the presence of uniform magnetic field has been considered by Finlayson [13]. Thermoconvective

stability of ferrofluids without considering buoyancy effects has been investigated by Lalas and

Carmi [14], whereas Shliomis [15] analyzed the linearized relation for magnetized perturbed

quantities at the limit of instability. Schwab et al. [16] investigated experimentally the

Finlayson’s problem in the case of a strong magnetic field and detected the onset of convection

by plotting the Nusselt number versus the Rayleigh number. Then, the critical Rayleigh number

corresponds to a discontinuity in the slope. Later, Stiles and Kagan [17] examined the

experimental problem reported by Schwab et al. [16] and generalized the Finlayson’s model

assuming that under a strong magnetic field, the rotational viscosity augments the shear viscosity.

Venktasubramanim and Kaloni [18] have studied the Bénard problem for a ferromagnetic

fluid, in a rotating layer. Their analysis is a linearized one which takes in to account oscillatory

convection. Zahn and Green [19] and Zahn and Pioch [20] examined instability problems where

the magnetic field has the effect of rendering the viscosity to be essentially zero or negative

depending on the field strength. The thermal convection in a layer of magnetic fluid confined in a

two-dimensional cylindrical geometry has been studied by Lange [21]. Shivakumara et al. [22]

investigated the effect of changing the steady temperature profile on thermal convection in a

ferrofluid. Odenbach [23] has given a comprehensive description of magnetoviscous effects in

ferrofluids in his monograph.

1.5 HYDRODYNAMIC AND HYDROMAGNETIC STABILITY

Stability can be defined as the quality of being immune to small disturbances. Thus, by

stability we mean permanent type of equilibrium state. For an equilibrium state or a steady flow

to be of permanent type, it must not only satisfy the mechanical equations but also be stable

against arbitrary small perturbations. We consider a hydrodynamic or hydromagnetic system in a

stationary state i.e. one in which none of the variables defining the configuration is a function of

time. To investigate its stability we have to determine the reactions of the system to arbitrarily

small perturbations. If the perturbations gradually die down, the system is said to be stable. If the

7

perturbations grow with time i.e. the system reverts to its initial position, it is said to unstable. If

the system neither departs from its disturbed state nor tends to return to its initial position, the

system is said to be in neutral equilibrium. Further if at the onset of instability, there is an

oscillatory motion with growing amplitude; the instability is termed as overstability. Instability of

the system even for a single mode of disturbance will qualify the system to be unstable whereas

the system cannot be termed as stable unless it is stable with respect to every possible disturbance

to which it can be subjected.

Hydrodynamics as well as magnetohydrodynamics are both governed by non-linear

partial differential equations. No general method exists to solve these non-linear partial

differential equations. However, in spite of the complexity of the equations determining a fluid

flow, some simple patterns of flow (such as between parallel planes, or rotating cylinders) are

permitted as stationary solutions. These patterns of flow can, however, be realized only for

certain ranges of parameters characterizing them. They cannot be realized outside the ranges. The

reason for this lies in their inherent instability, i.e. in their inability to sustain themselves against

small perturbations to which every physical system is subjected upon. It is in the differentiation

of the stable from unstable patterns of permissible flows that the problems of hydrodynamic

stability originate.

Let us consider a hydrodynamic or hydromagnetic system in which the equations

governing it are in stationary state. Let jXXX ,......,, 21 be a set of parameters, which define the

system. These parameters include geometrical parameters such as the characteristic dimensions

of the system, parameters characterizing the velocity field prevailing in the system, magnitudes of

forces acting on the system such as pressure gradient, temperature gradient, magnetic fields,

rotation and others. While considering the stability of such a system, with a given set of

parameters jXXX ,......,, 21 , we essentially seek to determine the reaction of the system to small

disturbances. If all the initial states are classified as stable or unstable, according to the criteria

stated above, then in the space of parameters jXXX ,......,, 21 , the locus, which separates the two

classes of states defines the state of ‘marginal stability’ of the system. This definition implies that

a marginal state is a state of ‘neutral stability’. The locus of the marginal states in the

jXXX ,......,, 21 -space will be defined by an equation of the form

8

.0,......,, 21 jXXX (1.1)

The prime objective of the hydrodynamic stability theory is to determine the locus of

marginal states.

Again, in discussing the stability of a hydrodynamic or hydromagnetic system, it is often

convenient to suppose that all the perturbations of the system, except one, are kept constant while

the chosen one is continuously varied till a critical value for it is obtained and the system passes

from stability to instability. We then say that instability sets in at this value of the chosen

parameter when all others have their pre-assigned values.

The states of marginal stability can appear in two ways:

i) If the amplitudes of a small disturbance can grow or be damped aperiodically, the

transition from stability to instability takes place via a marginal state exhibiting a

stationary pattern of motions.

ii) If the amplitude of a small disturbance can grow or be damped by oscillations of

increasing or decreasing amplitude, the transition takes place via a marginal state

exhibiting oscillatory motions with a certain definite characteristic frequency. We

have different terminologies characterizing the two states.

In classifying marginal states into the two classes – stationary and oscillatory, we have

supposed that we are dealing with dissipative systems. In non-dissipative, conservative systems,

the situation is generally somewhat different. In these cases the stable states, when perturbed,

execute undamped oscillations with certain definite characteristic frequencies; while in the

unstable states small initial perturbations tend to grow exponentially with time; and the marginal

states themselves are stationary.

If at the onset of instability a stationary pattern of motions prevails, then one says that the

‘principle of exchange of stabilities’ is valid and that instability sets in as a stationary cellular

convection or secondary flow. On the other hand, if at the onset of instability oscillatory motions

prevail, then one says that one has a case of ‘overstability’.

Now, hydrodynamic or hydromagnetic stability has been recognized as one of the central

problem of fluid mechanics. Much work has been on hydrodynamic or hydromagnetic stability

because of its importance in engineering, in meteorology and oceanography, in aerodynamics, in

hydraulics, in geophysics (study of winds and marine currents), in astrophysics (formation of

9

stars from interstellar gas, formation of planetary systems) and in the quest for thermonuclear

fusion. The first major contribution to the study of hydrodynamic stability can be found in the

theoretical papers of Helmholtz [24]. Even earlier many scholars had certainly become aware of

the question but their efforts did not progress beyond the stage of description. For example, the

drawings of vortices by Leonardo-de Vinci (fifteenth century) and the experimental observations

of Hagen [25] deserve mention.

Twelve years after the discoveries of Helmholtz, Lord Rayleigh [26] developed a general

linear stability theory for inviscid plane parallel shear flow, which was mathematically calculated

and had intuitively sensible results, and the combined efforts of Reynolds [27], Kelvin [28-29],

and Rayleigh [30-38] produced a rich harvest of knowledge. Reynolds [27] predicted that

Reynolds number was a crude measure of the relative importance of inertial (non-linear) effects

relative to the viscous processes in determining the evolution of the flow. He discovered the first

experimental evidence of ‘sinuous’ motions in water and is generally credited for a first

description of random or ‘turbulent’ flow. He made use of the dimensional analysis and

discovered the all-important number which is called the ‘Reynolds number’ these days. He

pointed out that disorder begins when Reynolds number exceeds a critical value and that special

stresses must be taken into account.

The founder of hydrodynamic stability is Lord Rayleigh, who published a great number of

papers (as cited above) regarding profile and the instability of rotating flows between cylinders.

Early in the twentieth century, studies on hydrodynamic stability were connected with the Bénard

experiments on thermal convection in thin liquid layers. Around 1907, it was generally believed

that the existence of the critical Reynolds number could not be explained easily and that the

problem involved both the effect of the second derivative of the mean flow and of the viscous

forces. The key equation was arrived independently by Orr [39] and by Sommerfeld [40]. This

Orr-Sommerfeld equation remained unsolved for twenty-two years, until Tollmien [41]

calculated the first neutral eigenvalues and obtained a critical Reynolds number. The work of

Taylor [42] on vortices between concentric rotating cylinders was the principal and best-known

contribution. Indeed this was a dual effect where theory and experiment were matched

simultaneously. Jeffreys [43] demonstrated the mathematical equivalence of the two stability

problems of convection and flow between rotating cylinders. In fact, it was the application of

newer mathematical techniques that brought the initial success to Tollmien [41].

10

Soon, following the same track, Schlichting [44-50] made further evaluations of the

critical Reynolds number and amplification rates of disturbances. The improved mathematical

procedure used by Lin [51, 52] not only removed the controversial issue of stability of Poiseuille

flows, but also laid the basis for the general expansion of the stability analysis. Any additional

doubts with respect to this system were finally settled down by the first use of a digital computer

in hydrodynamical stabilities. Magnetic, gravitational and convective effects were examined by

Bénard [53] and further elaborated by Chandrasekhar [11]. The monograph of Lin [54] settled

many controversial questions that had been built over the years. The study of compressible flows

was started with the work of Lees [55] and continued by Dunn and Lin [56]. Finally, the theory

of non-linear processes was set up by Meksyn and Stuart [57]. Later, some simple non-linear

problems have been successfully treated by Fromm and Harlow [58]. This work used a totally

numerical method and demonstrated the usage of modern computers. Some other good works in

non-linear theory, which need mention are by Coles [59], Segel [60], Reynolds and Potter [61],

Kirchgässner and Sorger [62], Stewartson and Stuart [63] and Weissman [64] etc.

1.6 BOUSSINESQ APPROXIMATION

Boussinesq approximation has been used in the Rayleigh discussion, because, in solving

the hydrodynamic equations we have difficulties regarding their non-linear character and the

variable nature of the various coefficients due to variations in temperature. Due to these

complications it is extremely difficult to solve these equations. So there is a need for introducing

some mathematical approximation to simplify the basic equations. Boussinesq [65] got rid of

various coefficient variations by taking them to be constants by applying some approximations

which are given below. However, non-linearity of equations still prevails under these

approximations. Boussinesq [65] first pointed out that there are many situations of practical

occurrence in which the basic equations can be simplified. These situations occur when the

variation in the density and different coefficients is due to variations in temperature of only

moderate amounts. The origin of simplification in these cases is due to the smallness of the

coefficient of volume expansion , whose range is 310 to 410 . For variations in temperature

not exceeding 010 C (say), the variations in density are almost one percent. The variations in

the other coefficients (consequent to the variations in density) must be of the same order. But

11

there is one important exception that the variability of ρ in the term of external force in the

equation of motion cannot be ignored. This is because the acceleration resulting from

ii TXαXδρ (where T is a measure of the variations in temperature which occur) can be

quite large. Accordingly, we may treat ρ as a constant in all terms in the equations of motion

except the one in the external force. This is the ‘Boussinesq approximation’. This

approximation makes the mathematics simpler and it has also gained a wide recognition in other

problems of non-homogeneous fluids, for example, the problems of Kelvin-Helmholtz instability.

Nevertheless, the equations which follow on the Boussinesq approximation are of interest in

themselves and they also provide the basis for further developments in the non-linear domain.

1.7 VARIOUS TYPES OF FLUIDS

A viscous fluid is a material continuum that is unable to withstand a static shear stress.

Such fluids have no surface tension. Flow of a viscous fluid at any moment is determined

completely by the shear forces acting on it at that moment. The greater the force, the faster will

be the rate of shear flow and the flow at zero force will also be zero. A viscous fluid can actually

be very rigid if it is of very high viscosity. Viscous fluids stay in the shape they have at the

instant that force is removed (they have no inertia). They can have arbitrary shape. Viscous fluids

in contact with each other do not coalesce. A desired portion of fluid can be moved without effort

(if moved slowly) into any location from any place else. Because it is viscous it can attach two

objects to each other (i.e., keep them in proximity). Broadly, viscous fluids can be classified as

Newtonian and non-Newtonian fluids.

1.7.1 NEWTONIAN AND NON- NEWTONIAN FLUIDS

Newtonian fluids are those fluids in which there is linear relationship between stress and

rate of strain. In other words, the stress components are linear functions of the rate of strain

components. The mathematical formulations of the physical assumptions that are taken to

characterize a medium are the constitutive equations (relation between stress and rate of strain).

The constitutive equation for an isotropic Newtonian fluid is

ijδeμeμτ kkijij

3

22 (1.2)

12

where μδeτ ijijij and,, are viscous stress tensor, rate of strain tensor, Kronecker delta and

coefficient of viscosity, respectively. This is the required relationship between viscous stress

tensor and rate of strain tensor.

All those fluids, which show non-linear relationship between stress and rate-of-strain i.e.

fluids that do not obey equation (1.2) are called non-Newtonian fluids. Fluids that cannot be

described by the Navier-Stokes equations are called non-Newtonian fluids. Further, if these fluids

possess elastic properties as well as viscous properties, then they are called viscoelastic fluids.

There is a growing importance of non-Newtonian fluids in geophysical fluid dynamics, chemical

technology and petroleum industry [Larson [66], Chin [67] and Khomami and Su [68]] The study

of convective fluid motion in porous medium has aroused the interest of many researchers

because of its important applications in prediction of groundwater movement, in atmospheric

physics, especially in petroleum industry, due to the recovery of crude oil from pores of storage

rocks. The studies for non-Newtonian fluids in this regard are also of interest in chemical

technology and industry. There is a vast variety of non-Newtonian fluids. Principal types of non-

Newtonian fluid include: Couple-stress fluids, viscoelastic fluids [Rivlin-Ericksen fluid, Walters’

(model B') fluid], Plastic solids, Power-law fluids, time-dependent etc.

1.7.2 RIVLIN-ERICKSEN FLUID

Let μμpxvδeτT ijijijijijijand,,,,,,, denote the stress tensor, shear stress tensor, rate-of-

strain tensor, Kronecker delta, velocity vector, position vector, isotropic pressure, viscosity and

viscoelasticity, respectively. The constitutive relations for the Rivlin-Ericksen viscoelastic fluid

are

,ijijij τδpT (1.3)

ijij et

μμτ

2 (1.4)

.2

1

i

j

j

iij

x

v

x

ve (1.5)

Rivlin and Ericksen [69] have proposed a theoretical model for such elasticoviscous

fluids. Such and other polymers are used in the manufacture of parts of spacecrafts, aeroplane,

13

tyres, belt conveyers, ropes, cushions, seats, foams, plastics, engineering equipments, adhesives,

contact lens etc. Recently, polymers are also used in agriculture, communication appliances and

in bio-medical applications.

Sharma and Kumar [70] have studied the thermal instability of layer of Rivlin-Ericksen

elastico-viscous fluid acted on by a uniform rotation. Sharma et al. [71] have studied the

thermosolutal convection in Rivlin−Ericksen fluid in porous medium in hydromagnetics. In

another study, Sharma et al. [72] have studied the thermosolutal convection in Rivlin−Ericksen

rotating fluid in porous medium. The thermal instability of Rivlin−Ericksen fluid in a porous

medium with relaxation and inertia in the presence of Hall effects has been studied by Sunil and

Singh [73].

1.7.3 WALTERS’ FLUID (MODEL B')

Walters’ [74] proposed another important kind of elastico-viscous fluid (model B') with

the constitutive relations

,ijijij τδpT (1.6)

,22 ijij et

μμτ

(1.7)

.2

1

i

j

j

i

ij

x

v

x

ve (1.8)

Such elastico-viscous fluids have relevance and importance in chemical technology and

industry. The effect of rotation, magnetic field and Hall currents on thermosolutal instability of

Walters’ (model B') fluid in porous medium have been studied separately by Sharma et al. [75,

76] and Sunil et al. [77, 78], respectively.

1.7.4 COUPLE-STRESS FLUID

Many of the flow problems in fluids with couple-stresses, discussed by Stokes, indicate

some possible experiments, which could be used for determining the material constants, and the

results are found to differ from those of Newtonian fluid. Couple-stresses are found to appear in

noticeable magnitudes in polymer solutions for force and couple-stresses. This theory is

14

developed in an effort to examine the simplest generalization of the classical theory, which would

allow polar effects. The constitutive equations proposed by Stokes [79] are

iiijkkij DμδDλpT 2 , (1.9)

sijskkijij Gερ

WηT

22 , , (1.10)

and ijijij ωηωηM 44 , (1.11)

where ijjiij VVD ,,

2

1 , ijji VVW ,,

2

1

, (1.12)

and jkijki Vεω ,

2

1 , (1.13)

where ρVεGωWDMTTT ijksiijijijijijij ,,,,,,,,,,

and ηημλ ,,, are stress tensor, symmetric

part of ijT , anti- symmetric part of ijT , the couple-stress tensor, deformation tensor, the vorticity

tensor, the vorticity vector, body couple, the alternating unit tensor, velocity field, the density and

material constants respectively. The dimensions of λ and μ are those of viscosity whereas the

dimensions of η and η are those of momentum.

With the growing importance of non-Newtonian fluids in modern technology and

industries, the investigations on such fluids are desirable. Since the long chain hyaluronic acid

molecules are found as additives in synovial fluid, Walicki and Walicka [80] modeled synovial

fluids as couple-stress fluid in human joints. The synovial fluid is the natural lubricant of joints of

the vertebrates. The synovial fluid is caused by the content of the hyaluronic acid, a fluid of high

viscosity, near to gel. Practically all diseases of joints are caused by or connected with a

malfunction of the lubrication.

Environmental pollution is the main cause of dust to enter into the human body. The

metal dust, which filters into the blood stream of those working near the furnace, causes

extensive damage to the chromosomes and the genetic mutations, so observed, are likely to breed

cancer or malformations in the coming progeny. Therefore, it is very essential to study the blood

flow with dust particles. Considering blood as couple-stress fluid and dust particles as micro-

organisms, Rathod and Thippeswamy [81] have studied the gravity flow of pulsatile blood,

15

through closed rectangular inclined channel with micro-organisms. Sharma et al. [82] and Sunil

et al. [83] investigated various instability problems of couple-stress fluid.

1.7.5 FERROMAGNETIC FLUIDS (FERROFLUIDS)

The two main features that distinguish ferrofluids from ordinary fluids are the polarization

force and the body couple. Magnetic fluids, also called “ferrofluids”, are electrically non-

conducting colloidal suspensions of solid ferromagnetic magnetite particles in a non-electrically

conducting carrier fluid like water, kerosene, hydrocarbon etc. A typical ferrofluid contains 2310

particles per cubic meter. These fluids behave as a homogeneous continuum and exhibit a variety

of interesting phenomena. These particles are coated with a stabilizing dispersing agent

(surfactant) which prevents particle agglomeration even when a strong magnetic field gradient is

applied to the ferromagnetic fluid. The resulting material behaves like a normal fluid except that

it can experience forces due to magnetic polarization. Ferromagnetic fluids are not found in

nature but are artificially synthesized.

Ferrofluids were discovered at the National Aeronautics and Space Administration

(NASA) Research Center in the 1960s. NASA scientists were investigating different possible

methods of controlling liquids in space. NASA scientists discovered that they could precisely

control the location of a ferrofluid through the application of a magnetic field. Additionally, by

varying the strength of the field, the fluids could be made to flow. As already mentioned, the field

of ferrofluid research is more than 40 years old. Thus it is clear that not only original publications

in journals or conferences have been released, but also textbooks have been published giving

overview on certain areas of investigation of these fluids. The famous book

“Ferrohydrodynamics” by Ronald Rosensweig [12] is still the standard textbook for people

entering the field of magnetic fluid research. Furthermore some other books on magnetic fluids

are available [see e.g. Bashtovoy et al. [84], Berkovsky et al. [85], Berkovsky and Bashtovoy

[86] and Blums et al. [87]]. The monographs by Berkovsky et al. [85] and Berkovsky and

Bashtovoy [86] give an extensive review of the application possibilities of magnetic fluids.

Fluids with clearly developed magnetic properties i.e. ferromagnetic fluids are objects of

interest for scientific research and have broad possibilities for practical use. Ferrofluids exhibit

unusual properties under an externally applied magnetic field, i.e. they can be confined,

positioned or controlled at desired places. In many lubrication situations it is required to place the

16

lubricant at a desired position and then to retain it there. Therefore, ferrofluids have been

successfully employed as lubricants in various hydrodynamically lubricated bearings. This

motivated several research workers to analyze ferrofluid lubrication for different bearing situation

under various simplifying assumptions. Chandra et al. [88], Kumar et al. [89, 90] and Sinha et al.

[91] presented mathematical analyses of ferrolubrication for various configurations using

Shliomis model. Ferrofluid technology is the basis of a wide variety of products used for high

technology applications in semiconductor and computer industries. They are the basis of

ingenious new techniques for separation of materials according to density. Jakabský et al. [92]

have studied the utilization of ferromagnetic fluids in mineral processing and water treatment.

They considered the utilization of ferromagnetic fluids as a separating medium and modifying

agent affecting the magnetic properties of the solid and liquid materials. Other commercial uses

are ink jet printing and magneto gravimetric preparations of nonferrous metals. One important

field of application of magnetic fluids that should be mentioned is their use in biomedical

applications. For example, their use as a contrast medium in x-ray examinations [Papisov et al.

[93]] or for positioning tamponade for retinal detachment in eye-surgery [Dailey et al. [94]] has

been reported. In medicine, ferromagnetic fluids are used for the determination of blood flow and

circulation, for blocking arterial aneurisms. Such a fluid can be introduced beneath the skin in the

neighborhood of tissue invaded by a tumor and the invading tissue then destroyed by laser

radiation. Furthermore, the application of magnetic fluids for the purpose of cancer treatment

either by hyperthermia [Chan et al. [95], Jordan et al. [96], Hergt et al. [97] and Hiergeist et al.

[98]], using the change of magnetization in an AC-field to heat the tissue, or by drug targeting

[Ruuge and Rusetski [99]], are obviously challenging possibilities. In summary we can say that

magnetic fluids are a unique class of material. Ferrofluid technology is well established and

capable of solving a wide variety of technical problems. There are many successful applications

of this engineering material and there is immense future potential.

1.7.6 MICROPOLAR FLUIDS

A general theory of micropolar fluids has been presented by Eringen [100]. These fluids

have such internal structures in which coupling between the spin of each particle and the

macroscopic velocity field is taken into account. Compared to the classical Newtonian fluids,

micropolar fluids are characterized by two supplementary variables, i.e., the spin, responsible for

17

micro-rotations and the micro-inertia tensor describing the distributions of atoms and molecules

inside the fluid elements in addition to the velocity vector. Liquid crystals, colloidal fluids,

polymeric suspension, animal blood, etc. are few examples of micropolar fluids. Kazakia and

Ariman [101] and Eringen [102] extended this theory of structure continue to account for the

thermal effects. Datta and Sastry [103] investigated the Bénard problem in the micropolar fluid

using the theory due to Eringen. They considered a layer of micropolar fluid between two

horizontal planes placed at a finite distance apart and this layer was heated from below in such a

way that the negative temperature gradient was maintained in the upward direction. They have

shown that the plot of Rayleigh number versus wave number has two branches separating the

zones of stability. In case of negative Rayleigh number, they observed the instability in the

system. Veronis [104] studied the thermosolutal convection in a layer of fluid heated from below

and subjected to stable solute gradient, the solute being salt. In connection with instability of

micropolar fluids, one may look on the papers by (Perez-Garcia and Rubi [105], Siddheshwar and

Pranesh [106, 107], Ahmadi [108], Sharma and Kumar [109, 110, 111 and 112], Reena and Rana

[113,114], Abraham [115], Rani and Tomar [116, 117], Dragomirescu [118] including several

others.

1.8 METHODS DETERMINING STABILITY

Generally, the stability of a system is determined by the following methods:

1.8.1 PERTURBATION METHOD

This is the most suitable method for establishing instability of a system. In this method,

the hydrodynamic system whose instability we wish to establish, is supposed to undergo a

specific, small trial displacement and the effect of the additional forces brought into play is

considered. If these forces thus produced tend to increase the displacement, thereby enhancing

the deformation of the system still further, the system is unstable.

1.8.2 ENERGY METHOD

The more general, widely used and the oldest method to discuss stability is the energy

method which can accommodate finite disturbances also. This method is based on the principle

of energy. It was first used by Rayleigh [119] in the calculation of the frequencies of vibrating

18

systems. This method can be found in the works of Reynolds [120], Orr [39], Stuart [121], Serrin

[122] and Joseph [123] etc.

In a mechanical system for which there exists a potential energy functionV , a stationary

state of the system will be unstable or stable according as V is strictly maximum or minimum.

For such a system (negligible dissipative forces) VT constant, where T denotes the kinetic

energy of the system. Suppose that V attains a strict minimum

0V for a stationary configuration.

When the system is disturbed,

0VV in all the neighbouring configurations.

Since

00 VTVT , where

0T is the initial kinetic energy of the system generated by

the small disturbance, we obtain

000 i.e. TTVVTT (1.14)

The system, therefore, remains in the proximity of the stationary configuration and does

not tend to deviate further from it. Hence the system is stable.

If

0V denotes a strict maximum for a stationary configuration, then

0TT and the

system will tend to depart further and further from its initial state. Hence the system is unstable.

Therefore, to find the stability of a configuration by this method, we must calculate the change

0VV in the potential energy of the system when it is given a small displacement satisfying the

boundary conditions. If this change is positive for all possible infinitesimal displacements, then

the system is stable whereas if this change is negative for any one particular trial displacement,

then the system is unstable.

1.8.3 NORMAL MODE ANALYSIS METHOD

Normal mode analysis method is used to determine the stability of the stationary state of a

hydrodynamic or a hydromagnetic system. This method is quite general and has found extensive

applications. The beauty of this method is that it gives complete information about instability

including the rate of growth of any unstable perturbation. Chandrasekhar [11] has used this

method throughout in his book “Hydrodynamic and Hydromagnetic Stability” while discussing

the various instability problems.

We start from an initial flow which represents a stationary state of the system. We assume

that the various physical variables describing the flow suffer small infinitesimal perturbations and

19

obtain the equations governing these perturbations. Also we make use of the linear theory by

retaining only linear terms in the equations governing perturbations. To study these equations, we

assume further that perturbed quantities have time variations proportional tonte . The parameter n

is, in general, a function of k (the wave number) and of other parameters defining the system.

With the dependence on time separated in this manner, the perturbation equations will involve n

as a parameter and solution of these last equations must be sought which satisfy certain boundary

conditions. The equation which involves n as a parameter is known as dispersion relation. If the

value of n determined by the dispersion relation is:

i. real and negative, the system is stable;

ii. real and positive, the system is unstable;

iii. complex, say, ir innn , where rn and in are real, then we have following cases:

a) If rn < 0, the system is stable;

b) If rn > 0, the system is unstable;

c) If rn = 0, the modes are oscillatory;

iv. Further, if rn = 0 implies that in = 0, then the stationary (cellular) pattern of flow prevails

on the onset of instability. In other words, “Principle of exchange of stabilities” is valid.

v. If rn = 0 does not imply that in = 0, then overstability occurs.

From this it follows that if n is real, then n = 0 will separate the stable and unstable modes

and we will always have exchange of stabilities.

1.9 SCOPE OF METHODS

The main drawback of normal mode analysis method is that in some problems the

dispersion relation becomes so much complicated that it is not possible to draw any meaningful

conclusions from it. Finally, normal mode analysis is based on the linearized stability theory and

therefore it has all the defects of linear theory.

Due to these drawbacks in the normal mode analysis, people have preferred, many times,

the energy method to determine the stability of a system. This method is based on the principle of

energy and hence it is more general method to attack the stability problems. In fact, this is the

oldest method of the stability analysis which can accommodate finite disturbances also. But this

20

method also is not without limitations. For example, it gives criteria sufficient for stability and is

silent about instability, and in some cases (like Couette flow) it gives too conservative results.

However, it is often used because of its peculiarities and physical implications.

The formulation of mathematics of energy method is rigorous and relatively simple. But

this does not diminish its capacity for truth. However the potentials of the linear theory of

stability and of the energy method are complementary. The linear theory of hydrodynamic

stability has the drawback that one cannot, in general, make judgment regarding the growth

potential of finite disturbances. It cannot be claimed with certainty that a given system will

remain stable if disturbed under conditions judged favorable by linear theory. This and other

questions about the effects of finite disturbances are in the domain of non-linear theory. A review

has been given by Drazin and Reid [124]. Comparison of stability limits as given by energy and

linear theory yields the range of values of relevant stability parameters in which sub-critical

instabilities (which means that the system first becomes unstable to steady finite amplitude before

it becomes unstable to infinitesimal disturbances) of the hydrodynamic system are possible.

Joseph and Shir [125] pointed out that energy method provides mathematically rigorous and

sometimes physically precise theory of sub-critical convective instability. There are cases, for

example, plane Couette flow, where both the theories cannot depict real state of affairs as

obtained experimentally. Energy theory errs in giving a safe Reynolds number, which is far too

conservative, and linear theory errs in giving the flow as always stable. However, there are cases,

as in case of Bénard problem in which the Boussinesq approximation is made and fluid is heated

from below, where both the theories coincide.

The energy method has not the potential of the linear theory for fine discrimination of the

limits of stability. The potential of energy method has not been fully utilized and it needs more

exploration in its field. The normal mode analysis method is quite general to determine the

stability of a system.

Nevertheless, in the present thesis, we shall confine ourselves to the normal mode

analysis and shall not be making use of the energy method. In the present time, the normal mode

analysis is most commonly used and has advantage over other methods as it gives complete

information about instability including the rate of growth of any unstable perturbation.

21

1.10 SOME INSTABILITY PROBLEMS

1.10.1 THERMAL INSTABILITY (OR BÉNARD PROBLEM)

Consider a horizontal layer of fluid of uniform density, which is subjected to an adverse

temperature gradient by heating it from below. Then the fluid at the bottom becomes lighter than

the fluid at the top and thus it becomes a top-heavy arrangement, which is potentially unstable.

As a consequence of this, there will be natural tendency on the part of the fluid to redistribute

itself to make up the deficiency in the arrangement. But this redistribution is prevented to a

certain extent by its own viscosity and therefore instability can set in only when the adverse

temperature gradient exceeds certain critical value.

The origin of the problem of the onset of thermal instability in liquid layers heated from

below lies in the experimental works of Bénard [126]. He carried out his experiment on a very

thin layer of non-volatile liquid (1 mm in depth), placed on a carefully leveled metallic plate

maintained at a constant temperature. The upper surface of layer was kept in contact with the free

air. It was found that the layer resolved itself into a number of cells, known as Bénard cells. The

principal facts established by the experiments of Bénard and others may be summarized as:

i. A certain critical adverse temperature gradient must be exceeded before the instability

sets in.

ii. The motions that follow on exceeding the critical adverse temperature gradient have a

stationary cellular convection.

The formation of this cellular convection takes place in two phases. The first phase is

quite rapid, lasting for a second or two for less viscous liquids like alcohol. For heavy oils,

especially when the upward flux of heat is small, this phase may be characterized by “semi-

regular regime”. The cells are nearly identical taking form of approximately regular convex

polygons. The second phase is, however, of permanent nature. Experimentally it is found that it is

difficult to maintain a constant flow of heat, but if one succeeds in doing so with extreme care,

the cells take the form of identical regular hexagons.

On earlier occasions, Rumford [127] and Thompson [128] have recognized the

phenomenon of thermal convection. The instability of Bénard model has been a subject of

interest till today and an excellent review of this work up to 1957 with special reference to its

possible fields of application has been given by Ostrach [129]. For mathematical details one may

22

be referred to “Hydrodynamic and Hydromagnetic Stability.” by S. Chandrasekhar [11],

Saltzmann [130] and Spiegel [131].

It was not so easy to find a mathematical theory, which could give a correct interpretation

of these experimental facts. As many as sixteen years were lapsed after Bénard’s experiments

when Rayleigh [37] could succeed in laying down the theoretical foundations of the subject for

the first time with his pioneering paper dealing with cellular convection in a fluid heated

uniformly from below. Rayleigh showed that there is a non-dimensional number that represents

the physical factors entering the problem. It is now called the Rayleigh number and is given by

the expression.

νκ

dαβgR

4

(1.15)

Here R denotes the Rayleigh number, g the acceleration due to gravity, α the coefficient of

volume expansion, dz

dTβ the uniform adverse temperature gradient which is maintained,

d the depth of the fluid layer, κ the thermal diffusivity and ν the kinematic viscosity. Rayleigh

further showed that instability must set in when R exceeds a certain critical value cR and

stationary pattern of motions must prevail when R just exceeds cR .

Jeffreys’ [132, 133] discussed the theoretical aspects of the Bénard problem and modified

the Rayleigh criterion for the number of boundary conditions. Pellew and Southwell [134] gave

intermediate steps and also confirmed that oscillatory motions are always damped whereas non-

oscillatory motions are always manifested. Chandrasekhar [135] has studied the Bénard problem

as a characteristic value problem and has determined the critical Rayleigh number

νκ

dαβgR c

c

4

(1.16)

which yields the critical adverse temperature gradient at which the thermal instability sets in.

Here cβ is the critical adverse temperature gradient.

Chandrasekhar [136] has reconsidered the Bénard problem in the presence of vertical

magnetic field and has obtained the critical Rayleigh number and the corresponding wave

numbers of unstable modes at marginal stability in the three cases, namely, boundary surfaces

both free, both rigid and lower boundary rigid and upper boundary free.

23

Boussinesq approximation has been used in the Rayleigh discussion. Because, in solving

the hydrodynamic equations we have difficulties regarding their non-linear character and the

variable nature of the various coefficients with the variation in temperature. Due to these

difficulties, there is a need for introducing some mathematical approximation to simplify the

basic equations. One of the contributions of Boussinesq [65] in these problems of thermal

instability is precisely at this point in the form of an approximation, which is after his name.

According to this “we may treat r as a constant in all terms in the equations of motion except the

one in the external force”. This is the “Boussinesq approximation”. This approximation has also

gained a wide recognition in other problems of non-homogeneous fluids, for example, the

problems of Kelvin-Helmholtz instability type. Nevertheless, the equations, which follow on the

Boussinesq approximation, are of interest in themselves; and they provide also the basis for

further developments in the non-linear domain.

1.10.2 THERMOSOLUTAL INSTABILITY (DOUBLE-DIFFUSIVE CONVECTION)

In classical thermal instability problems, it has been assumed that the driving density

differences are produced by the spatial variation of single diffusing property i.e. heat. It has been

shown that a new phenomenon occurs when the simultaneous presence of two or more

components with different diffusivities is considered. This problem has been probed, when we

think about ocean where both heat and salt (or some dissolved substances) are important. This

problem has been termed as ‘thermosolutal convection’ (or thermohaline convection). In these

problems the solute is commonly, but not necessarily, as salt. Related effects have now been

observed in other contexts, and the name double-diffusive convection has been used to cover this

wide range of phenomena. The problem of thermohaline convection in a layer of fluid heated

from below and subjected to a stable salinity gradient has been studied by Veronis [104]. The

physics is quite similar to Veronis [104] thermohaline configuration, in the stellar case where

helium acts like salt in raising the density but in diffusing more slowly than heat. Nield [137] has

studied the problem of thermohaline convection in a horizontal layer of viscous fluid heated from

below and salted from above. The thermosolutal instability in Rivlin-Ericksen fluid in porous

medium in hydromagnetics in the presence of Hall currents and rotation, separately have been

studied by Sunil et al. [138] and Sharma et al. [139]. In another study, Sunil et al. [78] have

studied the effect of Hall currents on thermosolutal instability of Walters’ fluid in porous

24

medium. This problem of the onset of thermal instability in the presence of a solute gradient is of

great importance because of its application to oceanography, atmospheric physics and

astrophysics. The heat and solute being two diffusing components, double-diffusive convection is

the general term dealing with such phenomena. Linear calculations have also been done for a

variety of boundary conditions by Nield [137] and for an unbounded fluid by Walin [140]. A

study of the onset of convection in a layer of sugar-solution, with a stabilizing concentration

gradient, when the layer is heated from below, has been made by Shirtcliffe [141]. He found that

the first stage of the development of convection layers similar to those described by Turner and

Stommel [142] is the appearance in a thin bottom layer of a cellular oscillatory motion, which

initially has a very definite period. When the solute gradient is stabilizing, Veronis [104] and

Sani [143] have found that finite amplitude sub-critical instability (convection occurs for finite

amplitude at a thermal Rayleigh number less than that given by the linear theory) is possible.

1.10.3 RAYLEIGH-TAYLOR INSTABILITY

Rayleigh-Taylor instability arises from the character of equilibrium of an incompressible

heavy fluid of variable density (i.e. of a heterogeneous fluid). The simplest, nevertheless

important, example demonstrating the Rayleigh-Taylor instability is, when we consider two

fluids of different densities superposed one over the other (or accelerated towards each other), the

instability of the plane interface between the two fluids, if it occurs, is known as Rayleigh-Taylor

instability. Rayleigh [144] was first to investigate the character of equilibrium of an inviscid,

non-heat conducting as well as incompressible heavy fluid of variable density which is

continuously stratified in the vertical direction. The two cases, first, two uniform fluids of

different densities superposed one over the other and second, an exponentially varying density

distribution, were also treated by him. The main result in all such cases is that the configuration is

stable or unstable with respect to infinitesimal small perturbation accordingly whether the higher

density fluid underlies or overlies the lower density fluid.

Taylor [145] carried out this theoretical investigation further and studied the instability of

liquid surfaces when accelerated in a direction perpendicular to their planes. The experimental

demonstration of the development of the Rayleigh-Taylor instability is described by Lewis [146].

Chandrasekhar [147] studied the effect of variable viscosity upon the above problem and

established that if the original density stratification is monotonically increasing upwards

25

everywhere in the flow domain, then there cannot exist any oscillatory modes. He also showed

that a variational procedure for solving the characteristic values is possible.

The most detailed consideration of the effects arising from surface tension is due to Reid

[148]. Surface tension is generally found to have a stabilizing effect on the Rayleigh-Taylor

instability. It was also proved that the wave numbers, which are stabilized by surface tension, are

independent of viscosity.

Kruskal and Schwarzschild [149] considered the effect of a horizontal magnetic field on

the development of the Rayleigh-Taylor instability. They established the stabilizing nature of

magnetic field for all perturbations except those at right angles to the magnetic field. Gupta [150]

has investigated the stability of horizontal layer of a perfectly conducting fluid, with continuous

density and viscosity stratifications. He has shown that contrary to the usual role of viscosity as a

damping factor, it may some times act as a destabilizing agent. Sharma [151] considered the

effect of rotation and a general oblique magnetic field on the Rayleigh-Taylor instability. A novel

form of MHD Rayleigh-Taylor instability has been studied by Robinson [152]. Sharma and

Sharma [153] investigated the effects of suspended particles on the Rayleigh-Taylor instability of

two superposed conducting fluids.

1.10.4 KELVIN-HELMHOLTZ INSTABILITY

The Kelvin-Helmholtz instability occurs when we consider the character of the

equilibrium of a stratified heterogeneous fluid in which the different layers are in relative motion.

The most important case is when two superposed fluids flow one over the other with a relative

horizontal velocity, the instability of plane interface between the two fluids when it occurs in this

instance, is known as “Kelvin-Helmholtz instability”. Helmholtz [24] and Kelvin [28] were

primarily interested in the stability of superposed fluids in a state of differential streaming. The

experimental observation of the Kelvin-Helmholtz instability has been given by Francis [154].

Some other fundamental works in this field of knowledge are those of Taylor [155], Goldstein

[156], Dyson [157], Case [158] and Howard [159]. Chandrasekhar [11] described the effect of

rotation on the development of Kelvin-Helmholtz instability and showed that it is least

uninhibited for perturbations in the direction of streaming. The stability of the interface between

the fluids in relative motion for incompressible and compressible fluids has been reviewed by

Gerwin [160]. Drazin [161] considered the Kelvin-Helmholtz instability of a slowly varying

26

flow. He considered the model of instability when air is blown over water in a wide long channel.

Such problems are important in many applications as it is rare in practice that a flow is both

steady and depends on one space co-ordinate only. The non-linear development of the Kelvin-

Helmholtz instability has been studied by Drazin [162], Nayfeh and Saric [163, 164], Weissman

[64] and many others. Sharma and Kumari [165] have studied the hydromagnetic instability of

streaming fluids in porous medium. In another study, Sharma et al. [166] have considered the

instability of streaming Rivlin-Ericksen viscoelastic fluid in porous medium.

1.11 VARIOUS PARAMETERS EFFECTING STABILITY

1.11.1 COMPRESSIBILITY

Fluids are divided into two categories. Those which undergo appreciable variations in

density and volume under the impressed forces fall under the category of 'compressible fluids'.

Then, there are those which undergo no noticeable changes in density and volume during motion.

They are termed 'incompressible fluids'. Compressibility is thus a measure of the change in

density and consequently, the change in the volume of a fluid under the effect of external forces.

The main practical categories of motion for which fluid compressibility plays a crucial role are:

i. Wave propagation within the fluid.

ii. Steady flow in which the fluid speed is of the same order of magnitude as the speed of sound.

iii. Convection driven by body forces, e.g., gravity, acting on fluid subject to thermal expansion.

iv. Large-scale convection of gases in the presence of body forces.

When the fluids are compressible, the equations governing the system become quite

complicated. To simplify them, Boussinesq tried to justify the approximation for compressible

fluids when the density variations arise principally from thermal effects. Spiegel and Veronis

[167] have simplified the set of equations governing the flow of compressible fluids under the

following assumptions:

(a) The depth of the fluid layer is much less than the scale height, and

(b) The fluctuations in temperature, density and pressure, introduced due to motion, do not

exceed their total static variations.

Under the above approximations, the flow equations are the same as those for

incompressible fluids, except that the static temperature gradient is replaced by its excess over the

27

adiabatic one and specific heat at constant volume, vC is replaced by specific heat at constant

pressure, .pC Within the framework of the linear theory, the compressibility effects are found to

promote the stability of the perturbed basic flow.

1.11.2 SUSPENDED PARTICLES

Various studies in viscous, viscoelastic, couple-stress fluids with dust particles, have

appeared in the literature [see for example Saffman [168], Scanlon and Segel [169], Sharma et al.

[170], Palaniswamy and Purushotham [171], Sharma and Sharma [172], Sharma et al. [173] and

Sunil et al. [174]] because of the importance of dusty fluids in a wide range of areas of technical

importance such as fluidication, environmental pollution and weather forecasting etc. The

influence of dust particles on viscoelastic flows has a great importance in petroleum industry,

pulp and paper technology, in the purification of crude oil and several geophysical situations. The

present study of dust particles can serve as a theoretical support for experimental investigations

e.g. evaluating the influence of impurifications in fluids like couple-stress fluid and

ferromagnetic fluid on thermal convection phenomena.

1.11.3 MAGNETIC FIELD

Consider a fluid to be electrically conducting and be under the influence of a magnetic

field. The electrical conductivity of the fluid and the prevalence of magnetic fields contribute to

two kinds of effects. First, by the motion of the electrically conducting fluid across the magnetic

lines of force, electric currents are generated and the associated magnetic fields contribute to

changes in the existing fields and second, the fact that the fluid elements carrying currents

traversing across magnetic lines of forces contributes to additional forces acting on the fluid

elements. Therefore in MHD the body force acting on the fluid is the Lorentz force that arises

when electric current flows at an angle to the direction of an impressed magnetic field.

The effect of a magnetic field on the stability of a flow in porous media is of interest in

geophysics, particularly in the study of the Earth’s core, where the Earth’s mantle, which consists

of conducting fluid, behaves like a porous medium that can become convectively unstable as a

result of differential diffusion. So we have considered the effect of variable horizontal magnetic

field on the linear stability of stratified Walters’ (model B΄) fluid in stratified porous medium in

one problem.

28

1.11.4 ROTATION

The physical aspect of convection in a rotating fluid layer is the driving force for analysis.

Since the inhibiting effect of rotation on the instability of a fluid layer heated from below has

been recognized as a phenomenon of major importance in Bénard convection since long. As

convection in a rotating system is relevant to many geophysical applications and to industrial

applications such as semiconductor crystal growing, it is not surprising that there have been many

articles dealing with theoretical or experimental analysis of this problem [see for example,

Chandrasekhar [175], Chandrasekhar and Elbert [176], Veronis [177, 178, 179], Roberts [7],

Rossby [180], Roberts and Stewartson [181], Chandrasekhar [11], Kloeden and Wells [182] and

Langlois [183]].

Thermal convection in a rotating layer of a porous medium saturated by a homogeneous

fluid is a subject of practical interest for its applications in engineering. Among the applications

in engineering disciples one can find the food process industry, chemical process industry,

solidification and centrifugal casting of metals and rotating machinery. More detailed discussions

of applications of thermal convection in porous media and particularly in rotating porous

domains are presented by Nield and Bejan [184].

1.11.5 HALL EFFECT (HALL CURRENTS)

Hall, a graduate student at John Hopkins University discovered the “Hall effect” in 1879.

Hall’s original experiments were limited to solid-metallic conductors. A thin, flat strip of width b

and thickness d was traversed by a current I. Two fine wires were connected at equipotential

points on opposite edges of the strip and in turn joined to the terminals of a sensitive

galvanometer. When the magnetic field H, was introduced at right angles to the face of the strip,

the galvanometer gave a steady deflection. The voltage indicated by the galvanometer is known

as the Hall voltage and is directly proportional to both current and magnetic field.

It is a well known fact that if the mean free path is much larger than the electron Larmor

radius, electrons will be able to gyrate freely round the magnetic lines of force several times

before suffering collisions. Consequently, the electrons and ions appear to be tied with the lines

of force in a way and this reduces their mobility transverse to the magnetic field; the whole

current will not flow along the electric field. This tendency of electric current to flow across an

electric field in the presence of a magnetic field is called “Hall effect”. The Hall effect is more

29

pronounced in the strong magnetic field or in the case of ionized gas (degree of ionization is

small). It has a dual effect on the stability like viscosity and magnetic field. In some

circumstances, it does not affect the stability.

Ware [185] included this effect in his study of stability waves in magnetically confined

plasma. Sato [186] and Tani [187] have considered the Hall effect in an incompressible viscous

flow of an ionized gas with tensor conductivities in channels. They found that the inclusion of

Hall currents gives rise to cross flow, i.e. a flow at right angles to the primary flow in a channel

in the presence of a transverse magnetic field. Taylor [188] pointed out that the Hall current has a

strong stabilizing effect on the low density plasma. Coppi [189] pointed out that as long as the

resistivity of the fluid is neglected; Hall term has no effect on the stability, which was further

confirmed by Buti et al. [190]. Hosking [191] pointed out that it has a destabilizing influence on

the stability of the system. Tasso and Schram [192] discussed in the context of macroscopic

theory the instability and stability effects and concluded that Hall effect has a destabilizing effect.

Gupta [193] has shown that the Hall currents have a destabilizing effect on the thermal instability

of a horizontal layer of a conducting fluid in the presence of a uniform magnetic field. He showed

that Hall currents induce a vertical component of vorticity and this may be the reason for

destabilizing influence.

The effect of magnetic field on the flow of plasma at a constant pressure gradient is

weakened as the Hall parameter increases. This is due to the decrease in the conductivity in the

direction of the induced electric field with an increase in the Hall parameter. The study of these

effects is of great importance because of its application to the physics of atmosphere and

astrophysics. In particular, Hall effects are likely to be present in the case of ionosphere and outer

layers of the solar atmosphere. Sharma and Sunil [194] have studied the effect of Hall currents on

thermosolutal instability of partially-ionized plasma in porous medium. The Hall effect on

thermal instability of Rivlin-Ericksen fluid has been studied by Sharma et al. [195]. The effect of

Hall currents on thermal instability of Walters’ (model B') fluid in porous medium has been

studied by Sunil and Kumar [77].

1.11.6 FLOW THROUGH POROUS MEDIA

Flow through porous media is a topic encountered in many branches of engineering and

science like ground water hydrology, reservoir engineering, soil science, soil mechanics and

30

chemical engineering. It is now appropriate to begin by presenting the concept of porous media,

porosity and permeability as encountered in practice.

1.11.6.1 POROUS MEDIUM

Initially, we may attempt to describe a porous medium as a “solid with holes.” Obviously,

a hollow metal cylinder would not normally be classed as a porous medium, nor would a solid

block with isolated holes or pores, since we seek to define a porous medium in connection with

flow through the medium, and not, for example, in connection with thermal insulation. We might

try to improve our definition by stipulating that the pores are interconnected, with at least several

continuous paths from one side of the medium to the other, and by somehow specifying a better

distribution (in either a regular or random manner) of holes and paths over the entire porous

medium domain. With these preliminary remarks, we may try to define a porous medium as

[Bear et al. [196]]:

(a) A portion of space occupied by heterogeneous or multiphase matter. At least one of the

phases comprising this matter is not solid. They may be gaseous and/or liquid phases. The solid

phase is called the solid matrix. That space within the porous medium domain that is not part of

the solid matrix is referred to as void space (or pore space). The matrix of a porous medium is the

material in which the holes or pores are imbedded.

(b) The solid phase should be distributed throughout the porous medium within the domain

occupied by a porous medium. An essential characteristic of a porous medium is that the specific

surface of the solid matrix is relatively high. In many respects, this characteristic dictates the

behaviour of fluids in porous media. Another basic feature of a porous medium is that the various

openings comprising the void space are relatively narrow.

(c) At least some of the pores comprising the void space should be interconnected. The

interconnected pore space is sometimes termed the effective pore space. As far as flow through

porous media is concerned, unconnected pores may be considered as part of the solid matrix.

Certain portions of the interconnected pore space may, in fact, also be ineffective as far as flow

through the medium is concerned. For example, pores may be dead-end pores (or blind pores),

i.e., pores or channels with only a narrow single connection to the interconnected pore space so

that almost no flow occurs through them.

31

Summarizing, all the features described above we can say that by a porous medium we

mean a material consisting of a solid matrix with an interconnected void. The matrix of a porous

medium is the material in which the holes or pores are imbedded. We suppose that the solid

matrix is either rigid (the usual situation) or it undergoes small deformation. The

interconnectedness of the void (the pores) allows the flow of one or more fluids through the

material. In the simplest situation (“single-phase flow”) the void is saturated by a single fluid. In

“two-phase flow” a liquid and a gas share the void space. The manner in which the holes are

imbedded, how they are interconnected and the description of their location, shape and

interconnection, characterize the porous medium. Accordingly, we have different classes of

porous media. Porous medium are classified as unconsolidated or consolidated and as ordered or

random. Examples of unconsolidated media are beach sand, glass beads, catalyst pellets, soil,

gravel etc. Examples of consolidated media are most of the naturally occurring rocks such as

sandstone, limestone and so forth. In addition concrete, cement, bricks, paper, cloth etc., are man-

made consolidated media. Ordered porous media are regular packing of various types of

materials such as spheres, column packing etc. Random media are media without any particular

correlating factor. In a natural porous medium the distribution of pores with respect to shape and

size is irregular. Examples of natural porous media are beach sand, sandstone, limestone, rye

bread and wood etc. There are two levels of description. At the microscopic level, the description

is statistical- in terms of “pore size” distribution. This description of “pore-size” is considered

nebulous because it depends on how one describes such a distribution. It is opposed to

macroscopic description that describes the media in terms of the average or bulk properties. Two

macroscopic properties of porous media which may be used to describe fluid flow are described

as follows:

1.11.6.2 POROSITY

The porosity ε of a porous medium is defined as the fraction of the total volume of the

medium that is occupied by void space. Thus ε1 is the fraction that is occupied by solid. For an

isotropic medium the “surface porosity” (i.e., the fraction of void area to total area of a typical

cross section) will normally be equal to ε . In defining ε in this way, we have assumed that whole

of void space is connected. If in fact we have to deal with a medium in which some of the pore

space is disconnected from the remainder, then we have to introduce an “effective porosity”

32

defined as the ratio of connected void to total volume. Porosity macroscopically characterizes the

effective pore volume of the medium. In homogeneous isotropic materials, e is a pure constant

but in non homogeneous materials ε may depend upon position. For man-made materials such as

metallic foams ε can approach the value 1 and in beds of packed spheres, ε is in the range of

0.25-0.50. For natural media, ε does not normally exceed 0.6.

1.11.6.3 PERMEABILITY

The conductance of the medium is defined with direct reference of Darcy’s law as the

seepage velocity of the percolating water per unit drop of the hydraulic head. The permeability is

related to pore-size distribution since the distribution of the sizes of entrances, exits and lengths

of the pore walls make up the major resistance to flow. The permeability is the single parameter

that reflects the conductance of a given pore structure. The dimensions of the permeability are

length squared. In oil industry it is measured in ‘darcy’ with 1 darcy = 291087.9 cm .

The permeability and porosity are related since if the porosity is zero the permeability is

zero. Although there may be correlation between porosity and permeability but permeability

cannot be predicted from porosity alone since we need additional parameters which provides

more information about pore structure.

When we consider flow in a porous medium we have to take into consideration some

additional complexities which are principally due to the interactions between the fluids and the

porous material. When a fluid permeates a porous medium, we cannot follow analytically the

actual path of an individual fluid particle because of the fluid-rock boundary conditions, which

must be considered. Thus in a porous medium one generally considers the fluid motion in terms

of volume or ensemble average of the motion of individual fluid elements over regions of space.

This is usually done by the famous Darcy’s [197] law and as a result of this the viscous term in

the equations of fluid motion will be replaced by the resistance term q

1k

μ , where μ is the

viscosity of the fluid, 1k the permeability of the medium and q the seepage velocity of the fluid.

A macroscopic equation which describes incompressible creeping flow of a Newtonian

fluid of viscosity μ through a macroscopically homogeneous and isotropic porous medium of

permeability 1k is the well-known Darcy’s equation.

33

p1

qk

(1.17)

where p is the interstitially average pressure within the porous medium and q is the filter

velocity (or Darcian velocity).

Lapwood [198], using Rayleigh’s procedure, has studied the convection in a fluid in a

porous medium and has shown that the criterion for the convective flow is 24πRc , where cR

is the critical Rayleigh number.

The study of convective fluid motion in porous medium has aroused the interest of many

researchers because of its important applications in prediction of groundwater movement and in

atmospheric physics, especially in petroleum industry, due to recovery of crude oil from pores of

storage rocks. In industrial applications, harmful particles can be filtered from a fluid stream by

passing it through a porous solid whose pores are too small to permit passage of particles.

Additionally, porous media may provide sites for chemical catalysis or absorption of components

of the fluid.

1.12 CONTRIBUTION OF THE PRESENT THESIS

The work embodied in the present thesis is divided into six chapters. Chapterwise

summary of work done is as follows:

1.12.1 CHAPTER-1

Chapter 1 is introductory. It reviews existing literature relevant to the thesis e.g.

hydrodynamics, hydromagnetics, ferrohydrodynamics, stability of the system, methods and

scopes determining stability etc. The thermal instability, thermosolutal instability and Rayleigh-

Taylor instability problems have been described and effects of various factors like Hall currents,

uniform/variable magnetic field, porous medium and viscoelasticity have been discussed.

1.12.2 CHAPTER-2

Chapter 2 is divided into two sections. In these two sections, the thermosolutal convection

in Rivlin-Ericksen fluid in porous medium is studied. In the first section, a layer of Rivlin-

Ericksen fluid heated and soluted from below in porous medium is considered in the presence of

uniform vertical magnetic field, rotation and suspended particles. It is found that for stationary

34

convection, the stable solute gradient and rotation have stabilizing effect on the system whereas

suspended particles have destabilizing effect. The medium permeability has a destabilizing effect

in the absence of rotation whereas in the presence of rotation it has a destabilizing/stabilizing

effect under certain conditions. The magnetic field has a stabilizing effect in the absence of

rotation whereas in the presence of rotation it has a stabilizing/destabilizing effect under certain

conditions. The critical Rayleigh numbers and wave numbers of the associated disturbances for

the onset of instability as stationary convection are obtained and the behaviour of various

parameters on critical Rayleigh numbers has been depicted graphically. The principle of

exchange of stabilities is satisfied in the absence of magnetic field, rotation and stable solute

gradient. The presence of magnetic field, rotation and stable solute gradient introduces oscillatory

modes into the system.

In the second section, the thermosolutal convection in Rivlin-Ericksen elastico viscous

fluid in porous medium is considered to include the effect of suspended particles in the presence

of uniform magnetic field, uniform rotation and variable gravity field. It is found that for

stationary convection, the stable solute gradient and rotation have stabilizing effect on the system

whereas suspended particles have destabilizing effect. The medium permeability has a

destabilizing effect in the absence of rotation whereas in the presence of rotation it has a

destabilizing/stabilizing effect under certain conditions. The magnetic field has a stabilizing

effect in the absence of rotation whereas in the presence of rotation it has a

stabilizing/destabilizing effect under certain conditions. The principle of exchange of stabilities is

satisfied in the absence of magnetic field, rotation and stable solute gradient. The presence of

magnetic field, rotation and stable solute gradient introduces oscillatory modes into the system.

1.12.3 CHAPTER-3

Chapter 3 is divided into two sections. In these two sections, the effect of Hall currents on

instability problems in ferromagnetic fluids is studied. First section deals with the theoretical

investigation of the effect of Hall current and suspended particles on the thermal stability of a

ferromagnetic fluid heated from below. For a fluid layer between two free boundaries, an exact

solution is obtained using a linearized stability theory and normal mode analysis. A dispersion

relation governing the effects of suspended particles and Hall current is derived. For the case of

stationary convection, it is found that the magnetic field has a stabilizing effect on the system, as

35

such its effect is to postpone the onset of thermal instability whereas the suspended particles and

Hall currents are found to hasten the onset of thermal instability, and as such they have

destabilizing effect on the system. The effects of various parameters on the thermal stability are

depicted graphically. The critical Rayleigh numbers and wave numbers of the associated

disturbances for the onset of instability as stationary convection are obtained and the behaviour of

various parameters on critical thermal Rayleigh numbers has been depicted graphically. The

principle of exchange of stabilities is not valid for the problem under consideration whereas in the

absence of Hall currents (hence magnetic field), it is valid under certain conditions.

Second section is the extension of the first section wherein the same problem has been

analyzed by including the effect of compressibility as additional parameters. For the case of

stationary convection, it is found that the compressibility and magnetic field has a stabilizing

effect on the system whereas the suspended particles and Hall currents are found to hasten the

onset of thermal instability.

1.12.4 CHAPTER-4

Chapter 4 is divided into two sections. In these two sections, the effect of hall currents on

ferromagnetic fluids in porous medium is considered. First section deals with the theoretical

investigation of the effect of Hall currents on the thermal stability of a ferromagnetic fluid heated

and soluted from below in porous medium. For the case of stationary convection, it is found that

the magnetic field and stable solute gradient have a stabilizing effect on the system, as such its

effect is to postpone the onset of thermal instability whereas Hall currents are found to hasten the

onset of thermal instability, as such it has destabilizing effect on the system. The medium

permeability hastens the onset of convection for all wave numbers .1 Mx The effects of

various parameters on the thermal stability are depicted graphically. The critical Rayleigh

numbers and wave numbers of the associated disturbances for the onset of instability as

stationary convection are obtained and the behaviour of various parameters on critical thermal

Rayleigh numbers has been depicted graphically. The principle of exchange of stabilities is not

valid for the problem under consideration whereas in the absence of stable solute gradient and

Hall currents (hence magnetic field), it is valid under certain conditions.

Second section deals with the theoretical investigation of the effect of Hall currents on the

thermal stability of a ferromagnetic fluid heated from below in porous medium in the presence of

36

horizontal magnetic field. For the case of stationary convection, it is found that the magnetic field

has a stabilizing effect on the system, as such its effect is to postpone the onset of thermal

instability whereas Hall currents are found to hasten the onset of thermal instability, as such it has

destabilizing effect on the system. The medium permeability hastens the onset of convection for

all wave numbers .cos1 2Mxx The effects of various parameters on the thermal stability

are depicted graphically. The principle of exchange of stabilities is not valid for the problem

under consideration whereas in the absence of stable solute gradient and Hall currents (hence

magnetic field), it is valid under certain conditions.

1.12.5 CHAPTER-5

Chapter 5 is divided into two sections. In these two sections, thermal stability of couple-

stress fluids is studied. In first section, the thermal stability of a couple-stress fluid in the

presence magnetic field and rotation is considered. For stationary convection, it is found that

rotation has stabilizing effect always whereas magnetic field and couple-stress have a stabilizing

effect under certain conditions. In the absence of rotation, couple-stress and magnetic field have

stabilizing effect on the system. The critical Rayleigh numbers and wave numbers of the

associated disturbances for the onset of instability as stationary convection are obtained and the

behaviour of various parameters on critical thermal Rayleigh numbers has been depicted

graphically. It is found that principle of exchange of stabilities is satisfied in the absence of

magnetic field and rotation. In second section, the problem of first section has been extended to

include the effect of suspended particles.

1.12.6 CHAPTER-6

Chapter 6 introduces the micropolar fluid. In the present chapter thermal stability of

micropolar fluid layer heated from below in the presence of hall currents in porous medium has

been studied. For the case of stationary convection, the effect of various parameters like medium

permeability, magnetic field, Hall current, coupling parameter, micropolar coefficient and

micropolar heat micropolar heat conduction parameter has been analyzed and results are depicted

graphically also.