Before we get started with the subject on
wave mechanics, I just would like to take
some time to introduce you to the basics of
fluid mechanics. You have a number of textbooks,
internet facilities, where you have a number
of lectures, PDF files, chapter wise and I
am sure that all these things, all these material
would be quite helpful in understanding the
subject fluid mechanics. Before we get into
the wave mechanics, what I have tried to do
is just to bring out some of the essential,
most essential aspects of this subject, which
actually is needed for the bigger topic that
is the wave mechanics.
So, I start with the general aspect the behavior
of ocean waves or the behavior of waves in
the ocean governs the driving forces responsible
for the different kinds of phenomena in the
marine environment. The different kinds of
phenomena that is starting with how the waves
are generated, how they propagate, how they
form, how they form and then how they propagate
and when they propagate, what are all the
kinds of deformations or phenomena it would
undergo in the absence or in the presence
of any obstructions. All these things will
be discussing later.
So, in general, in order to understand all
those phenomena the driving forces are very
important and the driving forces are governed
by the basic fluid mechanics. So, a variety
of structures are deployed or installed in
the ocean, the design of which needs an in-depth
knowledge on the characteristics of waves
which in turn require understanding of basics
of fluid mechanics.
So, this is accomplished by providing an overview
of fluid mechanics overview on the fluid mechanics
aspects. So, that it allows the reader to
understand the principles underlying in the
mechanics of ocean waves and it is motion
in deep as well as in coastal waters. So,
the purpose for going through recourse to
the fluid mechanics has been highlighted so
we get started.
We have three states of matter that exist
in nature, namely solid, gas and liquid. So
liquid and gas are termed as fluids, which
undergo deformation under the action of shear
stresses or shear forces. Most of them are
self-explanatory although they are a self-explanatory,
I just briefly discuss about the some of these
aspects. Now, the main distinction between
a liquid and a gas lies in the rate of change
of density, that is, the density of the gas
changes more rapidly or more readily than
that of a liquid.
So, that is, the rate at change, rate at which
the change of change in the density take place
that really governs, whether the fluid is
a liquid or it is a gas. It is a change of
a density of a fluid is negligible, then we
say that it is incompressible, it is defined
as incompressible.
So, let us go to the next slide. We basically
have two broadly classified fluids, types
of fluids. A fluid that has no viscosity,
no surface tension and is incompressible is
termed as ideal fluid. For such a fluid, no
resistance is encountered as it moves. Ideal
fluid does not change in nature however, fluid
with low viscosity such as that of such as
air, water however be treated as in ideal
fluid, which is reasonable and well accepted
assumption. A fluid that has viscosity, surface
tension and it is incompressible which do
exist in nature is termed as practical fluids
or real fluids. These are the basic definition
of a real fluid and an ideal fluids which
not very often, some of the students they
make a mistake of calling an ideal fluid as
ideal flow. Flow is not ideal; it is only
the fluid which is ideal. Flow can be irrotational,
rotational.
So next, History of fluid mechanics in brief:
There are a few books which has beautifully
described about the history of fluid mechanics
quite in detail and which is quite interesting
also, but what I have done is I have just
discussed in brief. The theories of a real
or real fluid of Euler and Bernoulli, these
are the scientist establish, they established
in the mid eighteenth century on these theories
included the conservation of law, laws of
for mass and momentum or kinetic energy. Then
this achievement was followed by considerable
progress towards the formulation of several
empirical formulae in the field of fluid mechanics
as well as in hydraulics.
Many of you might have heard the Navier and
Stocks. It was Navier; a French civil Engineer
during the early 19 th century found that
the real fluid theory does not yield a good
estimation of forces on structures due to
fluid flow. When the design of for a bridge
across river Seine was made so, this was realized
by Navier later and this allowed the viscous
flow theory in which a viscous term to the
momentum conservation equation was developed
by Navier in 1822.
I am not showing any equation, I am just giving
you some of them background. Then just few
years around the same time or may be 20 years
later it was Stokes from England who also
developed viscous flow theory. The momentum
conservation equation for describing the viscous
flow which is still a use is therefore, named
after them and is termed as Navier-Stokes
equation. Very often you see what is Navier
Stokes many of the equations, many of the
phenomena can be well describe by Navier-Stokes
equation.
Having gone through the basics of fluid mechanics,
very brief we will just look into the types
of flow. We have broadly classified eight
types of flows: Steady and Unsteady, Uniform
or Non-uniform, Rotational or Irrotational,
Laminar and Turbulent. Remember these are
flows what we had seen earlier is the fluid,
ideal and real fluid. Now, flows are given
here 8 forms, but there are some combinations
that are also possible.
So we will just examine one by one, the types
of flow, the steady and unsteady flow. Fluid
characteristics such as velocity, pressure,
density, temperature etcetera, at any point
do not change in time. For example, water
moving with a constant distance discharge
rate so, at any point when you consider the
flow of water with constant discharge rate
or in general that is only an example, in
general first, this kind of a flow at any
point x y z the rate of change with respect
to time, the rate of change of the property
which we are discussing theta u p or dou rho
with respect to time is zero that is do not
change with time.
If, it is a changing with time if, it changes
with time what is it? Unsteady flow. So, for
example, if it changes with time then for
example, here I have taken the behavior of
ocean waves. What do you mean by behavior?
Why ocean waves are unsteady? You will see
that later how a wave varies, In the case
of a regular wave, if you measure the waves
in the deep ocean so, this is the surface
elevation and you see that the surface elevation
is going to vary with respect to time. When
you measure the open ocean or if you later
you will also see that the waves are can be
this kind of a wave can approximately said
to be represented can be represented as a
sinusoidal wave.
So, these waves you see that when they are
moving on the surface it will have its particle
velocities and other characteristics that
will be changing within the fluid medium.
So, all those changes like the change with
the pressure etcetera or velocities will all
be changing with respect to time. So, that
is why we say that it is an unsteady flow.
Now, Uniform Flow: At any given instant of
time, when the fluid particles as stated earlier
u, pressure, density etcetera does not change
in both magnitude and direction from point
to point not from time to time from point
to point within the fluid, the flow is said
to be that is why we are showing it as dou
u by dou s. S is with respect to space. So,
the variation of velocity with respect to
space at any given of point of time is going
to be equal to 0.
So, Non-Uniform Flow: If the velocity changes,
if the velocity of fluid, when I say velocity
it also includes the other properties of the
fluid. If the velocity of fluid changes from
point to point at any given time instant,
then it is called as non-uniform flow the
ones which are having the red color font that
shows only the examples for the different
kind of flows. Here in the case of non-uniform
flow, flow of liquid under pressure through
long pipe lines of varying diameter. Suppose,
if the diameter is constant, if this diameter
is constant, the same flow of fluid under
pressure will be uniform flow.
So, now this is constant dia and now, we have
varying dia. That is going to take care of
whether the flow is going to be non-uniform
or uniform. Now, under steady or unsteady
and uniform or non-uniform, flow can exist
independently of each other that is you can
hither steady or unsteady, uniform or non-uniform,
but they can also have combinations combinational
are also possible.
So, for example, if you have a flow of liquid
under a constant rate and it is in a long
pipe line constant diameter. So, you see that
constant rate, constant diameter. So, here
I have said varying diameter, constant diameter.
So, if it is a constant rate in a long diameter,
but the flow is steady, I mean the diameter
is constant, then you have steady uniform
flow. Then the flow of liquid whatever I have
highlighted that is very important. So, the
flow of a liquid at a constant rate through
a conical pipe that is steady, because constant
rate is there so you have a constant rate
here, but the shape of the pipe is different.
So, the shape of that pipe dictates whether
the flow is going to be uniform or non-uniform
and the rate at which it is pump that is going
to be controlling whether the flow is going
to be steady or unsteady. So, you have a control
of the types of flow you want to generate.
So, with the changing rate of flow these cases
became become unsteady uniform and unsteady
become unsteady uniform, unsteady or non-uniform
flow all these cases are possible.
So, although this these are very fundamental
I have noticed that there is a kind of confusion
some of the students express when they are
asked for some question they fumble and they
get very easily confused between uniform,
steady, unsteady, non-uniform. So, please
remember the kinds of examples, the examples
which is if you remember the example that
will clearly indicate the type of flows etcetera.
You have what is meant by rotational flow:
A flow is said to be rotational, if the fluid
particles while moving in the direction of
flow rotate, about their mass centers that
is natural so it has to rotate about its mass
centers so, for example, liquid in a rotating
tank so, that is a classical example for anirritation[al]-
for a rotational flow very often we deal with
irrotational flow and some of the derivations
you will start the flow is ideal, fluid is
ideal, flow is irrotational. for example,
when we derive the velocity potential in order
to understand the wave mechanics that how
the wave propagates etcetera. we will be forced
to derive a velocity potential.
The basic assumption there is the fluid is
ideal, flow is irrotational. So, the fluid
particles while moving in the direction of
flow do not do not rotate about their mass
centers. This type of flow exist only in case
of ideal Fluid. Why ideal fluid? Only in ideal
fluid there is no tangential or Shear Stresses
that would occur. Although the subject is
quite simple you should remember all these
definitions forever.
So, it says that the flow this type of flow
that is irrotational flow exist only in the
case of an ideal fluid for which no tangential
or shear stresses occur, but if the viscosity
of the fluid is very small or very less or
does not have much of significance then in
that case even this kind of flow can be assume
even in the case of real fluids. For example,
when I told in the beginning air and water
are assumed as ideal fluid, because it has
very low significance, low viscosity etcetera.
So, it is only an assumption and the assumption
is quite acceptable. Now, for a fluid flow
to be irrotational, the following conditions
are to be satisfied. What are the following
conditions? That is it may easily be proved
that the rotational components about axes
parallel to y and x or x and y axes can be
are said here. dou y that is W x half into
similarly, for W y along the y axes and this
with respect to the x axes.
So, if at every point in a fluid which is
flowing, the rotational components as indicated
here W x equal to zero, W y is equal to zero,
and W z equal to zero and then automatically
you will get these expressions on the right
hand side. That is dou w by dou y equal to
dou v by dou z. So, if when you have a particle
motion or the velocities of the particle in
a fluid moving in the x direction and y direction.
So, you have u and v, u v and w then with
this using this differentiation etcetera this
relationship you will be in a position to
find out whether the flow with that kind of
velocities will that be following will that
can be can it be termed as an irrotational
flow or otherwise by simply trying to get
the dou this relationship through… So, u
v w are the velocities in the x y z directions.
Now, we move on to laminar flow. A flow is
said to be laminar if the fluid particles
move along straight parallel parts in layers
that is when you we surely see the particles
will be moving in layers. This can easily
simulated in a lab, by having a controlled
discharge over glass tube and this is the
classical example that is classical exercise
done in under graduate course.
So, you can see that if the fluid particles
are move along straight parallel path in layers,
such that the paths of the individual fluid
particles do not cross the neighboring particles.
So, each particle will be moving in layers
or in laminate in other words fluids, the
fluids appear to be moving or to move by sliding
laminations, something like sliding over.
So, this type of flow occurs when viscous
flows dominate the inertia force and particularly
at low velocities. So, what you do you have
a tube and you inject some die so, that you
can see the path and you control your reynold's
apparatus this is what I am trying to explain.
So you can just try to control your discharge
and if the discharge is very less and you
control it in such a way that you can easily
visualize the flow that is moving. In kind
of you can generate a kind of a laminar flow.
So, laminar flow can occur in flow through
pipes or open channels or even through porous
media. Of course, laminar flows are much easier
for us to solve compare to turbulent flows.
Turbulent Flows: This is just an opposite
to the laminar flow. A fluid motion is said
to be turbulent when the fluid particles move
in entirely random or disorderly manner, that
results in a rapid and continuous mixing of
fluid leading to momentum transfer as flow
occurs. See as far as this turbulent flow
is concerned, it has a kind of a direct connection
with our kind of studies in particularly when
it is a either wave mechanics or coastal engineering.
So for some of this turbulent flow you will
be a coming across I will we are discussing
well while I am describing about some of the
phenomena that we come across in both the
subjects like wave mechanics as well as in
the field of coastal engineering.
A distinguish characteristic of turbulence
is its irregularity, there is there being
no definite frequency, as in the case of wave
motion, and no observable pattern, as in the
case of large eddies. So, eddies are what
is this a different shapes and sizes are present
over large distances in such a fluid flow.
Flow in natural streams, artificial channels,
Sewers, etcetera.. are all few examples of
turbulent flow. It is quite difficult to deal
with turbulent flow compare to laminar flow,
because it is laminar flow is a kind of a
flow which can be termed whereas, turbulence
flow it is very difficult to term the type
of flow because it is going to be moving in
random.
So, having seen the types of flows in brief
we will just look at one of the important
equation that is the continuity equation and
continuity equation that is its speaks or
it tells about the conservation of mass. Now,
let me so I am considering a point A B C and
D. Now, this is A dash B dash C dash and D
dash and I am identifying an element a point
at this location at the center and my axes
is x and y and of course, you have the z.
Let me take this distance as delta x and this
will be delta y and I have delta z. So, my
point here is defined as P of x y z. Now,
when you consider an elementary rectangular
parallelepiped with sides as indicated, the
planes are also indicated here which we are
going to consider the flow is from left to
right and it starts at the center point. Consider
a plane running at the center point. So, through
this plane, at this point center point the
flow is taking place and the mass of the fluid
will be as indicated here.
So, now this as we consider that u v and w
are the velocities in x y z direction and
rho is the mass density so, the mass of the
fluid that is going to pass per unit time
through the phase normal to the x axis. At
point p will be rho into u into delta y and
delta z that is the area, this is the area.
Now, what is the mass? What is the then the
mass of the fluid that is flowing per unit
time into this through the phase A B C D.
So, we are coming we are talking about this
phase this phase will be the mass that is
rho u delta y and delta z that will be entering
but, what about the rate of change, rate of
change will be with respect to the x direction
that has to be added.
So, we add that as delta x, dou u dou by dou
x with respect to x. This is the mass, but
over what distance over? because the point
is on the in the center the distance will
be delta x divided by 2, but it will have
a negative sign because the reference point
which we have taken is a center and the phase
which we consider is the on the left hand
side, that is on the other side of the opposite
side of with that with that of the direction
of flow and that is the reason why we have
minus delta x by...
So, I write this as into plus, dou divided
by dou x. So, this will be the mass fluid
flowing per unit time through the phase A
B C D. Now, you consider you repeat the same
thing by considering this area, this phase,
this phase will be practically the same except
that except that because it is on the on the
side towards a positive direction of the flow
so you will have the same thing that is what
is written here.
So, you have a fluid flow per unit time in
this over this phase and over this phase is
that clear we have established that. Now,
having established that we need to get the
net mass of fluid that has remained in the
fluid medium, after all we are considering
this as a fluid medium and what is the net
mass of fluid that has remained in this fluid
medium per unit time through this pair of
phases, because we are taking only these two
pair of faces. So, that will be this minus
this then you will get as shown here.
But then delta x is not going to be a function
of x, because we have considered a an element
of pure fluid. When we consider an element
of fluid the sides are the lengths are fixed.
Hence the delta x 
can be removed out, but we retain the variation
of u and density with respect to space not
with respect to time. Here, with respect to
space, it is retained.
Now, by applying the same procedure the net
mass that remains through other two phases
can also be obtained instead of here we have
u so, you will have v and w that is all. So,
we consider this phase and this phase so,this
phase and this phase and similarly, the other
two phases that is this phase and the other
phase behind the board. So, by adding all
these three, we get the net total mass of
fluid that has remained in the cube per unit
time, which is going to be as indicated here
not indicated as calculated or as derived.
So, what does a conservation say conservation
law since the fluid is neither created nor
destroyed in the cube, any increase in the
mass of fluid contained in this space per
unit time, is equal to the net total mass
of fluid, that has remained in the cube per
unit. So, there is no creation or removal,
no addition or reduction or removal.
So, now the mass of the fluid in the cube
and its rate increase with respect to time,
the total mass that is you have rho into rho
t that is with respect to time and this is
going to be a mass. So, the mass of the fluid
in the cube and its rate of increase with
time can be easily expressed as shown here.
Now, we need to equate in order to have the
conservation laws satisfied, we have to equate
this with the earlier expression which is
now going to be written as this that is summation
of the net mass that has remained in the flow,
remained in the cube considering all the six
phases I mean three pair of phases.
So, now this is the final expression which
we have arrived now divide both sides of the
above expression by the volume. Once you do
that you are going to get taking element that
fluid medium shrinks to the point P, the entire
fluid medium that we have now expanded the
whole thing just to understand how this different
components act? now if you assume taking the
limits so, that the entire fluid medium shrinks
to point P the continuity equation is obtained
as we have seen earlier and this is going
to be the continuity equation.
This equation now represents the continuity
equation. In its most general form is applicable
for steady as well as unsteady flow, Uniform
flow, Non-Uniform flow, compressible as well
as incompressible. See, you the variation
with respect to time is considered there,
variation with respect to space is considered
there so, all these factors are included so
you can apply it for compressible as well
as for incompressible fluids and it is applicable
for all types of flow which we had discussed
so far.
So, in the case of steady flow, we know that
dou rho or dou rho by dou t has to be equal
to zero. So, the above equation then becomes
as shown here. So, this will be applicable
for both compressible and incompressible flow,
incompressible fluid but, but only if steady
flow is applicable, it can be applicable only
for so, now you remove for example, for an
incompressible fluid, the mass density will
not be changing with respect to x y z and
t and hence the above equation can be simplified
as dou u by dou x plus dou u by dou y and
dou w by dou z equal to zero.
So, this is the basic equation which is called
as a continuity equation which is widely used
for describing the flow in a medium. A force
consider acting on fluids in motion, that
different forces influencing the fluid motion
are due to gravity, pressure, viscosity, turbulence,
surface tension and compressibility and these
are all listed below. F g is the Gravity Force
due to weight as a fluid, is a product of
mass and gravitational constant then pressure
force due to pressure gradient, viscous force
due to the existence of viscosity then turbulence
force due to turbulence, surface tension force
due to surface tension and due to the elasticity
property of the fluid you can have the compressibility
force.
Now, if a certain mass of fluid in motion
is influenced by all the above mentioned forces,
all the above mentioned force components then
the Newton Second law of motion according
to which we can write mass into acceleration
will be the summation of all these force components.
Further, you have a respective components
in the x y z direction as we have seen here.
It is all the first one I mean x direction
then z direction then you have the y direction.
So, all other things a x, a y and a z are
the fluid accelerations in the x y z directions.
Now, in most fluid problem we, neglect the
elasticity force as a force to elasticity
and the surface tension. So, hence the equation
can reduce to as shown here gravity, pressure,
viscous and turbulent and this force can be
called as is known as the Reynold’s Equation
of Motion and for in the case of turbulence
flows, in the case of laminar flows, turbulence,
effect of turbulence are neglected. So, that
would result into the summation of gravity,
pressure and viscous force and this is called
as the Navier Stokes equation which is widely
adopted.
Then in the case of ideal fluid, then viscous
force is neglected and hence you have only
the summation of gravity forces and the pressure
forces resulting in what is called as the
Euler’s Equation of Motion. Which will try
to derive in the next class probabily.
