I will come to another very fascinating example;
just to illustrate the kind of importance
that fluid dynamics may have not just in medical
diagnostics, but also in medical
treatment in a combine package of diagnostics
and treatment.
So, we can think of like,
injection for sucking blood, for testing the
blood sample.
For example for testing for
sugar level in a diabetes patient, and then
transferring insulin to the same person based
on the prevailing level of blood sugar.
So, this is the very common procedure that
many patients have to undergo throughout
their life, and it is not a very comfortable
process.
So, one of the alternatives that one can
think of is instead of a traditional needle
one can think of a micro needle, very small
needle.
And the typical micro needle may be designed
by mimicking the act of the
mosquitoes bite.
This is called as biomimetics.
This biomimetics it does not mean that
we just copy what is there in nature, it is
impossible to copy what is there in nature.
But
we can get some lessons out of it, for example;
when a mosquito sucks blood it typically
creates a suction pressure or negative pressure;
that draws blood in to its mouth part.
So, you mimicking the above, the sucking action
in a micro needle may be provided by a
micro electromechanical pump and it can draw
the blood, very small volume of blood.
Then there can be a testing of the blood,
let us for example say a metal oxide based
semi
conductor or MOSFET not a metal oxide semi
conductor, but a MOSFET based blood
glucose sensor.
And then based on that we can immediately
get a result, that what is the
amount of glucose that is there in the blood
sample?
So, that the mosquito bite sensor
gives that answer.
And then based on that there can be smart
insulin delivery system, and
this entire process can be built in a package
which looks like a wrist watch which is
shown in this view .
This is just to say that one can have small
needle and the needle really can make sure
that
you can have a very smart painless testing
of blood sample to get the amount of glucose
and deliver the insulin accordingly.
How does it work?
One of course is like creation of
the suction pressure, but the design of the
micro needle is based on the fact that in
the
micro scale.
In fact, mosquitos’ labium is also of micro
meter scale, like typical 25
micron to 50 micron diameter.
And in the typical micro meter scale surface
tension
works beautifully; there are certain forces
which are not that important in the large
scale,
but may be important in the small scale and
surface tension is one such force.
So, because of the surface tension working
beautifully the droplet of blood which is
sucked from the bottom of the scheme can be
transmitted is easily with a very little
indentation force.
And that makes the device to work in a painless
manner.
Fluid mechanics is often amazing.
So, I can go on giving you examples, but I
just do not
want to like overburden you with examples.
I just want to let you make you feel that
fluid dynamic is not just the traditional
automobiles or aircrafts or power plants or
process plants that we can think of, but fluid
dynamics is just in all aspects of modern
science and technology.
And it is often amazing, because many times
it contradicts
common intuition.
Like rough surfaces may reduce frictional
resistances against fluid flow instead of
acting
as hindrances.
Without friction birds cannot fly and fish
cannot swim.
Symmetric
problems may have asymmetric solutions.
Presence of particulate inclusions in a flow
may reduce effective viscous nature of the
fluid.
A highly viscous flow may be a good
simulator of ideal flows with zero viscosity.
And time dependence of a flow depends on
the choice of reference frame.
Like, you cannot say whether the flow is steady
or
unsteady until and unless you specify a reference
frame.
Shear force may vanish
although shear stress may exist.
So, these are certain very interesting phenomenal,
and many more which contradict
common intuition.
And this is what is important.
Like, from my perspective what I can
share my own perspective or philosophy with
you.
That all of us are born with certain
intuition, like even if there is a very little
child who puts his or her finger in fire he
or she
knows that it will burn.
So, these are something which is intuitive
and this intuition is
correct.
But while going through experiencing in life
one understands that there are many
natural and physical phenomena which do not
go by intuition.
And then to get an
explanation to that to me that is the proper
learning of science.
So, I can give you the non-intuitive example,
that if you have rough surfaces; the rough
surface is supposed to create more hindrance
against fluid flow, but under certain cases
it
can be shown that the rough surface may reduce
friction, not explicitly but implicitly.
What it can do, that it will have a rough
hydro phobic surface in a small confinement
then this surface can give rise to small bubbles;
nanometers scale bubbles.
And the liquid
which is flowing on the rough surface is not
directly feeling the effect of rough wall,
what it 
is doing it is gliding on the cushion layer
of the bubbles.
So, we can say that is the rough that makes
it the smooth, because the roughness of the
surface is one of the key factor that has
triggered the formation of this nanoscale
bubbles.
And the water, the liquid water that is moving
on the bubbles this is just flowing in an
apparently frictionless manner because it
is not interfacing with the rough surfaces
directly.
So, studying fluid mechanics we can give a
perspective.
Although, this is primarily a
theoretical course, but we will have several
video demonstrations to make it like a virtual
experimental environment.
But we will be mostly discussing on theory
and experiments
and theory need to go together for us to learn
fluid mechanics.
And from the various
examples that I have illustrated or my emphasis
is that like fluid mechanist can really be
used to understand not just fundamental scientific
issues, but to help towards the
betterment of the.
So, with this little bit of introductory remark
we will move on to an issue which we want
to discuss before discussing what is the fluid.
That in fluid mechanics the initial
discussion will typically always starts with
what is the fluid.
It is a very involved
question, but it is also important to understand
that many times we have an intuitive idea
of what is the fluid.
But, like before that we will try to see that
even if we know what is
the fluid question is; what is the perspective
in which we are going to analyze it, analyze
the motion of it.
To come in to more concrete terms we will
consider a gas.
When we considered a gas we
are definite that like it is a fluid, because
there are certain substances which fall in
the
interface of a fluid and a solid.
So, we are not going in to liquids at this
moment and we
are just concentrating on gases, because all
of us agree that it is a fluid by the sense
that
like it conveys to us from a common sense.
Now, let us say that there is a container.
In this container there are some gas molecules;
question is that, how do we analyze this system?
One possibility is that we write the
equations of motions for each of these molecules.
When we say that we are interested to
write equations of motions for each of these
molecules think about the situation.
Each
molecule may have three translational degrees
of freedom and three rotational degrees of
freedom; that means 6 independent equations
for each molecule into the number of
molecules.
And the number of molecules; think of just
one mole and one mole really a
small quantity will have Avogadro number of
molecules.
So, think of a realistic system.
So, how many of unknowns you have?
And you will have
this number of matching equations of motion
and you have to solve for that to get a
physical picture of the molecular motion.
So, it is a fundamental way of analyzing the
motion and is known as molecular dynamics,
but one has to understand that it has
practical limitation that it cannot really
address a very large system.
It can address only a
small system with number of molecules not
significantly large.
Depending on the
computational resources, it may be thousands
or more, but it cannot be prohibitively
large.
So, what is the alternative?
There are a couple of alternatives.
One alternative is that;
instead of addressing individual molecules
you can make a statistical average of many
molecules.
So what you can do is, instead of directly
simulating the molecules you
statistically represent the group of molecules
by statistical properties.
And that is what is
commonly done in kinetic theory of gases.
So, in kinetic theory of gases what you do?
You address the behavior of gas statistically.
And it is because you do it statistically
you
really do not have to simulate individual
molecules in a real sense, you have to just
simulate the statistical behavior of molecules
in as took stochastic sense.
So that makes
the analysis computationally little bit more
convenient and that is known as Microscopic
Approach.
Now, we have to understand that microscopic
approach being convenient it may carry
some of its important implications.
For example, if you want to make measurement;
let
us say you want to make a measurement of pressure,
of a gas.
So, microscopic approach
really does not give you a clue of how to
go about that, instead of that you may have
a
more convenient approach: you just have a
device which measures the time average
normal force over a given area and divide
the force by the area to get what is known
as
pressure.
In the microscopic approach you will find
pressure because of as a
consequence of change in molecular momentum
as it encounters a collision.
But in a
macroscopic approach you just do not care
about all those, but you just find time average
force over a given area.
So, that is called as macroscopic approach.
If the macroscopic approach is working then
that is best for us, because then you can
create the fluid as a continuous medium disregarding
the discontinuities.
So, you can
think of that the fluid is like a continuous
medium and that is known as Continuum.
And
the hypothesis that tells that the fluid can
be considered as a continuous medium,
disregarding the discontinuities inside following
the macroscopic approach is known as
continuum hypothesis.
So question is, does the continuum hypothesis
always work or it may not work?
The
thing is that if the continuum hypothesis
works it is the most convenient to use, because
we can use well known rules of differential
calculus to calculate the gradients of
properties.
So, we can express the behavior through well
known differential equations of
fundamental physics, classical physics to
represent the property variations within the
fluid.
But the issue is that can we do it for all
cases.
Now to get a more detailed insight on that,
let us say that we are interested to calculate
the density of the gas.
To calculate the density of a gas what we
need to do?
We need to
basically identify a elemental volume, we
find out the number of molecules in that
elemental volume; let us say that small m
is the mass of each molecule.
So, this is the
total mass divided by volume.
So far so good, but how small the volume should
be?
To get a real point to point
variation this volume should be as small as
possible, but not tending to 0.
It can tend to a
critical volume up to which the continuum
hypothesis will be valid not below that.
Why not below that?
Because, then the interrogating volume may
really have a very few
number of molecules.
If it has a very few number of molecules then
what will happen,
then this molecules remember there in random
motion.
So, what is going to happen is
that let us say there two molecules and suddenly
one molecule is out of this, which is
which is very common thing that can happen.
Then it can give rise to an error like which
is like an 100 percent type of error that
it can give rise to; so 50 percent type of
error
depending on how you are measuring the error.
So, when you have this high percentage of
error then that means, that is because of
the
uncertainties in the molecular occupancy of
the chosen interrogating volume.
So, when
can that happen?
That can happen if the volume is very small
or the volume may not be
that small but the system has the few numbers
of molecules.
That is called as the rarefied
system.
So, we can understand that because of uncertainties
with regard to the number of
molecules, when it has a large number of molecules
its fine, but if a if the volume has too
large number of molecules then an if the molecules
itself is large to handle that then we
do not get point to point variation of properties.
So, what we really want that is the small
volume but that should contain sufficient
number of molecules.
And that means it is not
a rarefied system.
The next question comes, what is sufficient
number of molecules?
How many numbers
of molecules you say that it should be sufficient?
Or when do you say that the system is
large or the system is small?
When do you say that?
To understand that we will come in
to more quantitative terms, because smallness
or largeness is qualitative; if we say that
the system is small you may say that it is
small to you, but it is large to me.
So, it is
always important to make a quantitative assessment
of the smallness or largeness.
So, to
understand that what we can do is, we use
one of the important quantities which is
lambda.
What is lambda?
Lambda is the molecular mean free path.
Molecular mean free path is what?
Molecular mean free path is the average distance
that
a molecule will travels before encountering
a collision.
So, that is the molecular mean
free path.
Now a system is relatively rarefied if the
molecular mean free path is large;
that means there are few molecules so that
a molecule before encountering another
collision has to travels the large distance.
But large and small has compared to what.
So,
we did compare lambda with something called
as L, which is called as the characteristic
length scale of the system.
So, what is the characteristic length scale?
A characteristic length scale is a distance
over
which characteristic changes can takes place.
For example, like if you have flow of gas
through a pipe.
You can see that characteristic changes takes
place from the wall where
the velocity is 0 to the center line where
the velocity is maximum.
So the characteristic
length can be the radius of the pipe, but
in engineering typically it is considered
as the
diameter of the pipe with the understanding
that it does not change the order, like
diameter is just 2 times the radius.
So, if we compare lambda with L; if lambda
is large compared to L then we say that it
is
a rarefied system.
But if lambda is small as compared to L, we
say that it is not a rarefied
system.
So, it is not just the lambda that is important,
it is not just the L that is important,
but lambda by L is a very important parameter
that talks about the rarefication of the
system.
So, this is known as a non-dimensional number;
this is the ratio of two length so
it is not-dimensional this is called as Knudsen
numbers.
So, a small Knudsen number means the system
is not that rarefied and continuum
hypothesis can be used.
But, if the Knudsen number is large; that
means that the system
has relative rarefaction; that means that
continuum hypothesis cannot be used and one
has to go for either statistical approach
through microscopic approach or may be
molecular dynamic to analyze the problem.
So, to summarize what we can say is that.
There are several approaches; one is the
molecular dynamics approach to analyze the
fluid flow, and which is most intuitive but
computationally most challenging.
And there is a compromise; one can go for
statistically average behavior of many molecules
which is the statistical mechanics
approach.
And the most convenient is the macroscopic
approach based on continuum
view point, where we consider the fluid as
a continuous medium disregarding the
discontinuities.
And the continuum hypothesis can be used only
under a certain
conditions typically governed by this Knudsen
number.
So, if the continuum hypothesis
can used then it is very convenient, because
we can use the well known rules of
differential calculus for solving the problems.
And, because this is a very introductory course
we will be mostly dealing with fluid
dynamics where continuum hypothesis can be
safely used.
So, we will be encountering
situations and solving problems which we will
address through the use of continuum
hypothesis.
From the next lecture onwards we will continue
with the discussion with
which we are leaving today.
Thank you very much.
