Now, we have discussed about one fluid property
which is pressure; whenever we are
talking about effects of compressibility,
there are other related fluid property which
come
into the picture in a very related manner,
and those properties we will look into one
by
one briefly.
So, one of the important properties will be
density.
So, loosely what we say that if we
have a volume elemental volume say delta V,
we have the mass of the molecules which
are there in these delta V.
So, we take mass per unit volume all of us
like to write limits.
So, we write limit this as delta V tends to
0, we will think that nicely it should give
a
limiting definition of what is the so called
density.
Mathematically very nice we will see
whether it works or not.
So, what it says it says in the limit as delta
V tends to 0; that means, limitingly small
volume, you find out what is the mass of molecules
inside.
So, you find that mass per
unit volume to get local density at a point;
that is what this definition is saying.
Whether
it works or not we have to come back to the
continuum hypothesis to adjust.
So, if we
remember that in the continuum hypothesis,
we disregard the molecular nature and we
just considered it is a continuous medium,
does not mean that there are no molecules,
but; obviously, when abstracted off the molecules
we are just representing their gross
effect.
But whenever there are molecules we have to
see that what is the number of
molecules within this elemental volume; again
if the number of molecules is within this
elemental volume is very small, then because
of the statistical fluctuations even
uncertainty in 1 molecule will give a lot
of error, and will give a lot of fluctuation.
So, it is critical that what is that elemental
volume that it should choose, it cannot be
too
small.
What is the smallness?
The smallness will come with the length scale;
the
smallness of the length scale here is the
mean free path lambda.
So, a very small volume
when we say then that will scale with lambda
q, lambda is like a length scale which will
correspond to a very small volume.
So, when the volume is of the order of lambda
q
elemental volume, then it will have lots of
uncertainties in the statistical fluctuations
of
the molecules; because within that length
scale you really have uncertainties related
to
collision.
On the other hand if you take this volume
delta V very large, then also we can
calculate a density, but it will not be able
to capture the local variation, it will give
a
global average therefore, one has to choose
a threshold length scale for calculating this
density, and how it should be sensitive to
the length scale if you make a plot of say
the
length scale that we choose say we call it
L s and the density that we predict.
So, you will see that if you choose a very
small length scale you will get a variation,
this
type of fluctuation then it will come to a
steady one, and then if you choose a larger
length scale it will be changing like this.
So, what is the significance of such a plot;
these
length scales are small enough, so that we
have really random fluctuations because of
the
uncertainties.
This length scale is fine beyond this length
scale you have variation this
variation is because over the system length
scale the density is varying from one point
to
the other.
So, a correct choice of length scale may be
something which can be say in
between this 2; so in between these 2 limits.
Therefore if we say delta V tends to 0 that
is not fundamentally correct; because delta
V
tends to 0 may make you fall on these region,
because delta V tends to 0 means
mathematically delta V is as small as possible.
So, as small as possible will; obviously,
be something smaller than as big as possible.
So obviously, in that context one has to
remember that this delta V tends to 0 should
be corrected; and how it should be
corrected?
We should change it as not delta V tends to
0, but delta V tends to some delta
V star, which is like if we call this as L
s star than it may be it is of the L s star
q.
So, it is
a threshold length scale, beyond which you
are not having such uncertainties and
fluctuations affecting your density calculation.
So, this delta V star therefore, we can say
is the smallest elemental volume over which
continuum hypothesis is valid.
So, it is not tending to 0, but tending to
a limitingly small volume delta V star, over
which still continuum hypothesis works.
Below this limit continuum hypothesis might
not work and therefore, this definition will
not work because this definition is on the
basis of continuum description of fluid property
like density.
So, we have talked about
density we have talked about pressure next
let us talk about bulk modulus.
So, all of you are aware of the basic concept
of bulk modulus, but let us just see that
how
you have defined it.
Let us try to define it first in very loose
manner, it is always
important to get a qualitative field and then
of course, you can have more sophisticated
definition.
Will not go into very detailed sophisticated
definition of bulk modulus
because it requires a detail understanding
of thermodynamic processes and therefore,
we
are not going into that type of definition.
So, in a loose sense if you are applying say
a pressure differential delta P that is expected
to give rise to a change volume of a fluid
element; let us say that change in volume
is
delta V.
So, original volume was V.
So, this is the rate of or this is the total
change in
volume per unit volume.
So, this is the kind of volumetric change,
and this is the
pressure differential which is responsible
for the volumetric strength; and you expect
that
if delta P is positive, delta V is negative
because if you press a fluid element it should
compress it is volume should decrease.
If you want to give the corresponding fluid
property a positive number definition then
you should adjust it with a negative sign.
Now, you can relate the change on the volume
with a change in density; how is it
possible to relate the change in volume with
the change in density?
So, you have say
consider the mass of the fluid element, so
that is the density into volume.
From now
onwards whenever we will be discussing about
the volume we will be using a symbol not
V, but V with a strike through because we
will be using V for velocity also.
So, just to
avoid the confusion between the symbol of
velocity, and symbol for volume we will be
just distinguishing those in this way.
So, I will not be repeating the symbol many
times,
but once I am using this type of symbol, you
just take it that we are talking about the
volume not the velocity
So, if you want to say see that what is the
relationship between the elemental change
of
density with the elemental change of volume;
what you can do simply just take a log of
both sides and differentiate.
So, if you differentiate keeping in mind that
the mass of the
fluid element is conserved.
So, it is derivatives should be 0 therefore,
loosely like if you
are following this definition, we can relate
delta V by V with delta rho by rho.
So, that
will just absorb the minus sign, and it will
be like this.
So, it is like rho delta p over delta
rho like this.
Now, we can relate delta p with the velocity
of flow; in a order of magnitude sense the
delta p and velocity of flow this is just
like if you consider that there is a equivalent
pressure change, which is brought about by
the change in kinetic energy of a fluid, which
is moving with a velocity u, then this is
this is not that exactly equal it is just
to say that
one scales with the other in this way.
So, you can therefore, write a scale of K
as.
So, rho
let us try to see that what is delta rho by
rho scale; that means, what is the change
in
density related to it is original density.
So, if you do that it will be half rho U square
by
K.
Remember one thing that this K by rho, it
is something which is a very fundamental
quantity, which we have studied in physics
what is this or square root of K by rho if
it
reminds you more.
.
This is fundamentally sonic velocity sonic
speed so to say.
What is sonic speed?
Sonic
speed is not just speed of sound, sonic speed
is a speed by which a disturbance
propagates through a medium and here we are
talking about this type of disturbance to
the elastic property of the medium, so it
happens to the speed of sound.
So, this is the
sonic speed a.
So, where a we call as sonic speed, this is
a very basic high school physics
basic definition.
So, keeping this in view we can write this
as half u square by a square.
So, you can see that the relative change in
the density is related to a quantity u square
by
a square; what is this u square by a square?
This is a non-dimensional quantity that you
can see because it is a ratio of 2 velocities.
So, in the numerator you have u, in the denominator
you have a.
So, u is the velocity of
flow and a is the velocity of a disturbance
which is moving in the medium in which the
flow is occurring, and these 2 ratios is known
as Mach number, I mean ratios of this 2
numbers is known as Mach number.
So, you have heard about the Mach number like
a
jet moving with a Mach of this.
So, higher the Mach number higher is the velocity
of
flow related to the velocity of the disturbance
with which the disturbance propagates
within the medium and therefore, we say that
it is having a more and more compressible
effect.
The reason is if you have if you just write
this delta rho by rho, you see that it will
scale with half of square of the Mach number
therefore, higher the Mach number higher
is the effect of the change of density relative
to it is original density.
So, Mach number therefore, is a very important
indicator of something which is called as
compressibility of a fluid.
So, what is the next what is the signature
of compressibility of
a fluid, we will say that a fluid is compressible
when it has a change in density because
of a change in pressure.
So, in that way all fluids are compressible
right because all
fluids will have some change in density because
of change in pressure.
But when we say
that a fluid is incompressible what we mean
is that, that effect is negligibly small.
So, a
compressible fluid and an incompressible fluid
these are just conceptual paradise, there
is
no fluid as such which is incompressible,
but when we say that a fluid is incompressible
we mean that it is compressibility effect
is very very small; again how small or how
large
that is something which may be debated.
So, let us say that we are talking about a
change this relative change say 5 percent.
So, let
us say that if we say that this change is
less than 5 percent we say that it is almost
incompressible.
So, if you want to see that what will lead
to that 5 percent?
So, one may
work it out with say 5 percent means 5 divided
by 100.
So, what would be the threshold
Mach number for this roughly 0.3 right?
So, 0.33 or whatever roughly 0.3; that means,
if
we say that a relative density change less
than 5 percent is something which we do not
consider as a compressibility effect, that
implicitly means that a Mach number less than
0.3 is something, which is not going to give
us any serious compressibility effect.
So, this is important because whenever you
are analyzing an engineering flow, nobody
will tell you that whether the flow is compressible
or incompressible.
As an analyzer it is
your responsibility to make a judgment of
whether you are going to use the concept of
compressible flow or incompressible flow for
the analysis of your problem.
And then
you have to be confident that whether a particular
analysis methodology is going to work
or not of course, for all flows compressible
flow and analysis will work, because all
flows are compressible, but it is like if
you have a mosquito you will not like to kill
it
with a canon.
So, if you are ready if you are having the
possibility of doing the deleting
the simple analysis, you should not go for
a complex analysis.
That is what all of us are
learnt in engineering that do not go for unnecessary
complication until and unless it is
absolutely required.
So, whenever compressible flow analysis is
not required we should not go for it, and
this
Mach number of flow will give us a guideline
of whether we should go for a
compressible analysis or not.
A couple of other important points or remarks
are there
regarding this definition of bulk modulus,
one is see in this definition we have talked
about change in volume, because of a change
in pressure or equivalently change in
density because of a change in pressure.
But pressure effect of change of density it
depends on the type of process, all of you
have heard of certain thermodynamic
processes like adiabatic process, isothermal
process and so on.
So, given a particular
system how the density will change with pressure,
will depend on the nature of the
thermodynamic process.
So, this definition as such fundamental is
not incorrect, but incomplete; because it
does
not talk about the thermodynamic process by
which you are trying to have this change of
state.
So, there are more fundamental or correct
definitions of this in terms of specifying
it as say either reversible isothermal process
reversible adiabatic process and so on, but
we are not going into those details here because
thermodynamics is not the scope of this
particular course.
But we should keep it that in mind because
whenever you will be
starting thermodynamics, again this type of
definition will come into the picture and
there more detailing will be done in terms
of whether it is a reversible adiabatic process,
reversible isothermal process and so on.
So, that is one of the important concepts.
The second important concept is as follows;
say you are interested to identify whether
a
flow is incompressible or not, and in that
respect there is a subtle difference between
the
concept of incompressible fluid and incompressible
flow, these are very very subtle
concepts.
So, when you talk about an incompressible
flow what you mean is that if you
have a volume element of a fluid that volume
does not change.
So, incompressible flow
means that that there is no volumetric strength
of the fluid element, there is no change in
volume.
But you cannot directly always relate it with
this definition because the change
in volume may not always be due to change
in pressure directly, it may be because of
something else also.
And there are reasons for which you might
have change in volume
of a fluid element not because of the change
in density due to change in pressure, but
may be because of change in density due to
change in temperature, not directly due to
pressure.
So, whenever we are talking about incompressible
fluid, we are talking about that we are
asking ourselves a question that is there
a change in density because of a change in
pressure; if that answer is that yes it is
significant we call it a compressible fluid,
but not
compressible flow definition is something
more general, compressible flow means a
fluid element which if it is going to have
a volumetric strength or change in volume
per
unit volume by whatever reasons, it need not
be just due to pressure or it may be because
of anything, then we say that it is a compressible
flow.
So, compressible fluid and
compressible flow are related because of course,
one of the reasons of being a fluid
compressible or being a flow compressible
because the fluid itself is compressible,
so the
density change due to change in pressure is
significant, but there could be other effect
that has creating the change in volume.
So, this is the concept that we should remember.
Next what we will do is we will try to learn
about very important property of fluid,
which is called as viscosity.
So, when we talk about viscosity, we will
not try to just
learn it in abstraction, but we will start
with an example.
Let us say that you have a flat
plate just like a top of the table a flat
plate, and fluid is coming from fast stream
just like
say fluid is being blown from that side, it
is coming on the top of the table and going
away.
So, the top of the table will be like a flat
plate.
So, let us say that the fluid is coming with
the uniform velocity from a free stream; in
fluid mechanics usually we give such a symbol
infinity, with a subscript we indicate that
it is a free stream condition.
So, infinity subscript is like a free stream
velocity.
So, it is
coming with a uniform free stream velocity,
now that free nuance will disturb because
of
the present of the plate, and let us see that
what is going to happen.
So, when this fluid
first comes in contact with the plate what
happens?
Let us first try to understand that say
there is a fluid molecule which comes in contact
with a plate.
So, will the plate like to do
with the fluid molecule?
Let us consider 2 different examples one is
for a gas and
another is for a liquid; usually whenever
we discuss about fluids we are either talking
about gases or liquids, but sometimes their
physical behavior it is better to discuss
distinctly or differently.
So, let us say that there is a gas molecule
as a first example which is coming falling
on
this plate.
So, what will happen?
There will be first a tendency that the gas
molecule is
absorbed on the surface.
So, once it is absorbed on the surface then
what will happen that
it will exchange some of his momentum with
the surface, so it will try to have it slow
down, and then again the it will try to be
each get getting ejected from the surface.
So, it
is like a molecule falling on the surface
absorbed on the surface, getting ejected from
the
surface like this.
So, in this process many molecules are colliding
with this and they are exchanging their
momentum with the wall.
So, if there are very large number of collision.
So, to say
theoretically infinitely large number of collisions,
then this kind of momentum exchange
will bring on an average the molecules equilibrium
with the surface.
So, if the surface is
at rest the molecules will also be at rest.
So, that will imply that there is 0 relative
velocity between the fluid and the solid at
the point of contact, and this is something
which is known as no slip boundary condition.
So, fundamentally what is no slip
boundary condition?
It is 0 relative tangential component of velocity
to be more
accurate, 0 relative tangential component
of velocity between the fluid and the solid
at
their points of contact.
We are not talking about a normal component,
because still the molecule may be
colliding like it may have a sort of elastic
collision.
So, it may bounce back.
So, it may
have a normal component.
Now obviously, regarding the normal component
there are
issues like if the molecules are sufficiently
large in number and they are at the wall,
they
cannot penetrate and go through the wall,
wall is not having holes.
So, that is called as a
no penetration boundary condition then they
are even the normal component of velocity
will become 0; but no slip boundary condition
does not talk about that, that is the
separate consideration no slip boundary condition
talks only about the tangential
component of velocity.
So, 0 relative tangential component of velocity
between the fluid and the solid at the
point of contact.
Now as I am telling this to you, you are tending
to believe that this is
always the correct picture, and this has happen
really for a long time.
So, for a long time
this no slip boundary condition was taken
as something which is like a ritual which
we
should not change, and the reason was that
for many or for most engineering flows it
is
still valid, or it has been experimentally
found to be very very accurate.
But whenever we are understanding this concept
we should ask ourselves a question; are
their conditions in which the no slip boundary
condition may be highlighted.
It is
important because in many of the modern day
application of fluid mechanics, specially
fluid mechanics in small length scales this
boundary condition is something which is put
under serious question.
So obviously, we need to see that or we need
to appreciate that
this is just a conceptual paradigm, it is
not something which is a ritual and which
is
expected to work always; let us see let us
try to look into an example within the context
of gas flow that no slip boundary condition
does not work.
So, let us say that you have gas molecules,
but not very large number of gas molecules.
So, then what will happen?
The molecules will be exchanging momentum
with the wall,
but there will not be very large number of
collisions; because there will not be very
large
number of collisions the momentum exchange
will not be complete.
So, there will be
some velocity of the fluid relative to the
solid boundary, even if otherwise we tend
to
believe that there should not be any slip.
So, that is just because of the rarefied nature
of
the medium, that there is not sufficiently
large number of molecules to have a
theoretically large number of or infinite
number of collisions.
On the top of that there may be local strong
gradients in density and temperature, and
that might itself induce motion of molecules
of gases over the solid surface.
So, these are
called as phoretic motions.
If these are induce by temperature this are
called as
thermophoresis, and this may be induce by
any other effect, but temperature is one of
the
common effects by which by introducing a very
high gradient of temperature you can
introduce local flow of molecules of gases
over the solid boundary.
So, we can see that
there may be situations and there are likely
to be such situations when the no slip
boundary condition is not valid, but well
in most of the engineering system that we
are
talking about the no slip boundary condition
will work for gases.
Except for verified
gases or may be gases which are not having
sufficiently large number of molecules, or
gases being subjected to very high local gradients
of density or temperature; for liquids it
is difficult to believe that the no slip boundary
condition will not work; because liquid
bound liquids are very compact systems.
So, liquid molecules will not be insufficient
in number to have in inadequate collisions
with the wall, but for liquid molecules there
may be slip because of certain reasons.
So,
to understand the picture of the what happens
in the liquid molecules, let us consider a
small element of surface like this, the surface
may look very very smooth on the top, but
if you look it into a very powerful microscope
it will be much much worse than what I
have drawn here.
So, it will have lots of peaks and valleys,
and what will happen is that
the molecules will nicely seat on this peaks
and valleys.
So, some will be entrapped like
this, and because of a compact nature what
will happen whatever is entrapped is not
easily been escaped, and that will make us
believe that yes it will be a no slip boundary
condition.
At the same time if you are having a very
high shear rate which is being
introduced from the liquid, say a very high
rate of shear strength what will that try
to do
that will try to forcefully dig this out from
or take this out from this locations.
So, then in that kind of context the liquid
molecules may also slip on the surfaces;
otherwise if you have very smooth surfaces,
say you must have heard about carbon
nanotubes these days those are very sophisticated
and fascinated technologies to produce
carbon nanotubes.
So, those are very smooth tubes and if you
are having liquids in
contact with them now; obviously, there will
be the Vander wall forces of interaction
between the surface and the liquids, but if
in in such case water flowing through this
nanotubes will have very ordered hydrogen
bonding, and then the motion of that water
will be such that it can overcome the Vander
walls forces of interactions those are
relatively weak in comparison to this strong
bonding on in the water, to overcome the
wall attraction and flow on the top of such
surface.
So, it may actually slip and these are called
highly slipping surfaces therefore, we have
to
keep in mind that no slip boundary condition
is a paradigm, which will work for most of
the engineering problems that we are going
to consider, but at the same time we should
not take it as a ritual.
We should keep in mind that there are situations
in which it might
be violated, but for practical purposes for
almost all the problems that we are going
to
solve in this particular course no slip boundary
condition we will work.
So, let us stop here today and in the next
class we will take this up and introduce the
concept of viscosity through this no slip
boundary condition that we have discussed
today.
Thank you.
