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
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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
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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
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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.
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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
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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
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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
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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
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.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
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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].
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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
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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)
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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-