Which is a pretty important fluid property
in the context of our discussions and we will
try our best to understand it first qualitatively
and try to see that how we can mathematically
express the fluid flow behaviour in terms
of the fluid property viscosity.
We start with recalling the no slip boundary
condition that we were discussing in the previous
lecture.
So what was the consequence of the discussion
that we concluded that in many operations,
the paradigm of no slip, that is 0 relative
velocity between the fluid and the solid at
the points of contact, of course that means
the tangential velocity component, that particular
situation gives rise to a boundary condition
at the fluid-solid interface known as no slip
boundary condition.
We will try to see what is the consequence
of the no slip boundary condition.
So the way we see the consequence, we have
by this time understood that if the solid
boundary is stationary, then a fluid which
is coming from a far stream with a uniform
velocity say u infinity and encountering the
solid boundary, what it will first do?
It will first have a disturbance, the disturbance
is being imposed by the solid boundary.
So if we want to make a sketch of how the
velocity varies with height at any section,
say at this identified section.
At the wall, if the no slip boundary condition
is valid, then since the plate is stationery,
the fluid is also stationery.
So the fluid velocity is 0.
Next you consider a layer of fluid which is
just above this one.
This layer is subjected to 2 effects.
One is the effect of what is there at the
top of it and what is there at the bottom
of it.
At the bottom, there is a plate and there
is a stagnant layer of fluid molecules adjacent
to the plate and this stagnant layer does
not want the upper layer to move fast.
On the other hand, the fluid which is above
that layer, is not feeling the effect of the
wall directly.
So that is trying to make the fluid move faster;
therefore, it is being subjected to a competition
where the bottom layer is trying to make it
move slower, the upper layers are trying to
make it move faster and it has to adjust to
these.
So where from this adjustment comes.
If the bottom layer was not there, then perhaps
it would have not understood or failed the
effect of the wall.
Because what we are intuitively expecting
is that there is some property of the fluid
by virtue of which this message that there
is a wall gets propagated from the bottom
layer to the upper layers and there must be
some messenger for that and qualitatively
that messenger is through the fluid property
known as viscosity.
So viscosity is a kind of messenger for momentum
disturbance.
So this is a disturbance in the momentum of
the fluid.
So there must be some mechanism that plays
within the fluid by which a momentum disturbance
is propagated and because of this momentum
disturbance what happens?
Because of this momentum disturbance, there
is a resistance in relative motion between
various fluid layers.
So viscosity is also responsible for creating
a resistance between relative motion, against
relative motion between different fluid layers.
So let us see that how the relative motion
takes place.
So first you have at the wall 0 velocity,
then as you go up, you have a velocity higher
than this one.
It is not same as u infinity but it is definitely
better than 0 because it does not feel the
effect of the wall directly.
If the fluid has no viscosity, perhaps it
would never felt the effect of the wall, but
now because the fluid has viscosity, the effect
of momentum disturbance is being propagated
from the bottom layer to the top.
That is how this layer feels it, not directly
but what implicitly and accordingly it slows
itself down but as you go to higher and higher
positions, you see that the velocity is becoming
greater and greater and eventually it will
come to a stage when it reaches almost the
u infinity, the freestream condition.
So one of the important understandings is
that if you draw the locus of all these velocity
vectors, you can make a sketch which will
represent how the velocity is varying over
the section which is taken along this red
line.
And this type of sketch, we will encounter
many times in our course, is known as velocity
profile.
So it is giving a profile of variation of
how the velocity varies over a section.
This velocity profile comes to a state where
beyond which you really do not have any significant
variation in the velocity.
That is it has almost reached u infinity.
What does it mean?
It means that say beyond this if you go, these
have reached 99% of u infinity.
So beyond these if you go, it will be only
little change or for all practical purposes,
no change.
That means beyond these, the fluid does not
directly feel the effect of the wall.
It does not feel the effect of the wall at
all.
That does not mean that the fluid does not
have viscosity.
It has viscosity but the momentum disturbance
could propagate only up to this much.
So we can see that we may demarcate the physical
behaviour.
One is below this threshold location and another
above this one.
Below this, the fluids adjust itself with
the momentum disturbance.
Above this, it does not feel the momentum
disturbance.
Let us consider a second cross-section.
So let us say that we go to this cross-section,
another one a cross-section like this.
So when we want to plot a velocity profile,
at the wall because of no slip boundary condition,
it is 0 fine.
Now let us say that we are interested about
plotting the velocity at the same location
as this one.
So now you tell me whether it will be more
or less than what was here?
What should be the commonsense intuition?
Less, why do you feel that it should be less?
So it is like now more and more fluid is being
under direct effect of the plate, so there
is a greater tendency that the fluid is being
slowed down more and more.
When the fluid first enter, only a few fluid
elements were subjected to the effect of the
plate.
Now that more and more fluid has been subjected
to the effect of the plate, the effect of
slowing down is stronger.
So you expect that here the velocity will
be vb somewhat less than what it was here.
In this way in all sections it will be like
this.
What it will imply is that it will take a
greater height here to reach the almost the
u infinity because of a greater slowing down
effect.
So it may reach u infinity, say at a height
here.
It is not exactly u infinity but say 99% of
u infinity.
We are happy with that because for all practical
engineering purposes that is as good as u
infinity for us.
So again we may have a velocity profile here,
whatever and then we can make a very interesting
sketch.
What type of sketch?
See at every section, we are having a demarcation
between a position below which viscous effects
are strongly important and beyond which, these
effects are not so important.
So accordingly, we may draw a demarcating
line between these 2.
So when you consider this particular section,
you say that this is the location up to which
viscous effects are strongly creating gradients
in the velocity.
So to say here up to this much and so on.
So if you joint these with an imaginary line,
this is not that there is such a line in the
fluid but it is just a conceptual demarcating
boundary between a regional close to the wall
where viscous effects are very important.
And this region is thinner as this velocity
is higher.
We will see that later on that if this velocity
is very small, this region actually propagates
almost towards infinity but if this velocity
is quite high, of course there are other parameters
involved, we will see what those are but qualitatively
if this velocity is very high, this region
is thin and this region is changing, like
it is not of a fixed dimension.
So this imaginary line demarcates between
the outer region where viscous effects are
not important so to say and an inner region
where the viscous effects are important and
the inner region where this viscous effects
are important, that region is known as a boundary
layer.
So we will be discussing about the details
of the boundary layer through a separate chapter
later on but we are just trying to develop
a qualitative feel of what is a boundary layer
because it has a strong consequence with the
concept of viscosity.
And the other thing is that, this is a clever
way of looking into the problem.
If you want to analyse a problem where of
course the fluid is having some viscosity,
then outside the boundary layer, you may not
have to care for the viscosity.
So it is almost like a fluid without viscosity
as it is behaving because the momentum is
not further getting transmitted to create
a change in the velocity.
So if you have a viscous flow analysis within
this region, that may be good enough coupled
with an ideal flow analysis or fluid flow
analysis without viscosity outside this region.
So that is why conceptually this boundary
layer is a very important concept.
Not only that, most of the interesting physics
in the flow takes place within this layer.
And therefore it is very important to characterise
this particular behaviour.
So we have loosely seen the no slip boundary
condition and its consequence before we more
formally look into the viscosity.
Let us look into 1 or 2 animated situations
where we try to understand the implication
of the no slip boundary condition.
So let us look into that.
So this is going to be a representation through
a coloured die that what is the visual representation
of a no slip boundary condition.
So if you look into it carefully, you see
that if you focus your attention on the region
which is there are at the interface between
the fluid and the solid, the entire die was
concentrated on that, may be let me play that
again so that you can see it again.
So carefully see what happens on the surface.
You say that where the fluid almost tries
to adhere to the surface and at the end that
will be cleaned.
At the end of the movie, just to show that
there the fluid was almost like a stagnant
one because of the no slip boundary condition.
So to have different than maybe a more artificial
point of view, let us look into this.
So at the wall because of the no slip boundary
condition, you have the so-called particles
or molecules sticking but as you go outside,
you see that, as you go further and further,
you see that there is a velocity profile that
is being developed.
There are velocity gradients which are being
developed.
So this is a very important concept and when
we discuss about these concept in a greater
detail, maybe before that let us see another
one where you have 2 cases.
See qualitatively, we are trying to understand
what is the effect of viscosity, so the upper
panel represents the case with low viscosity
and the lower panel, represents a case with
high viscosity.
So if you see the case with low viscosity,
you can find out a very important demarcation
between the upper and the lower case.
Visibly what is the demarcation?
So the boundary layer for this so-called low
viscosity case is very thin, whereas for high
viscosity case, it is thicker that means what
the high viscosity case is trying to do?
It is trying to propagate the disturbance
of momentum imposed by the plate to a greater
distance, right.
So effect of viscosity is in terms of also
to the extent by which the effect in disturbance
of momentum is propagated into a medium.
Of course, we will go into the mathematical
quantification of this but my first intention
is that we first develop a qualitative feel
of or the physical feel of what we are going
to discuss about.
Now whenever we are going to discuss on these
concepts, obviously we will not always be
having a molecular picture and as you recall
in a continual hypothesis that if you consider
a molecule may be just like an isolated particle,
fluid is a collection of such isolated particles
and whenever we are going to discuss about
the behaviour of the fluid in terms of its
viscous nature here, we will be mostly bothering
on the continuum nature.
So let us just look into a sequence of animated
pictures to see that how you can have a transition
from a particle nature to a flow nature.
So this is the behaviour with 1 particle,
then maybe 2 particles isolated.
So these particles are like balls.
So these are idealisations.
Do not think that these are like real fluid
particles.
These are just to develop certain concepts.
So you see that you can have more number of
particles, isolated particles.
Then let us look into the next sequence of
images.
So now you are dropping those particles through
this funnel, okay.
This group behaviour is enforced by (()) (16:53).
Now let us see the next in this sequence.
So if you see a third example.
We look into fourth example straightaway.
Some problem with playing this.
So we will continue but if you just look into
the small part of the, small version of the
figure, you can appreciate that if you have
more and more number of particles very densely
populated together, you will see that it is
almost like as if there is a continuous flow
that is coming out of the funnel.
So depending on the compactness and the nature
of the particles, you may start with a particle
nature and at the end, you may come up with
a fluid flow nature.
So that is the whole hallmark of a continuum.
That in a continuum, we are basically looking
for a continuous distribution of matter and
we will first try to analyse the viscosity
behaviour through that continuum understanding.
So let us say that we take a fluid element.
So we take a fluid element at a particular
location and that particular location maybe
say close to the wall within the boundary
layer.
So let us say that we are taking this fluid
element which was originally rectangle.
Now let us see that what happens to the nature
of the fluid element or its geometrical characteristics
if it is subjected to viscous effects.
So what will happen?
To understand it, let us zoom its picture.
Originally let us say that name of this fluid
element is ABCD.
The layer AB has a tendency say to move with
a velocity u.
Let us assume that the velocities are all
along the positive x direction and therefore
since this moves with the velocity u, if we
consider a small time interval of delta t,
A will move to a position say A prime with
velocity of u so that the displacement is
u*delta t.
Now the upper layer, say it is moving with
a velocity of u+delta u.
Now we can understand that why it should be
different.
Because we have qualitatively discussed that
there is some effect which propagates the
disturbance in momentum through the fluid
and therefore these 2 layers are expected
to be of different velocities and let us say
that the fluid element that we have taken
is quite thin one with thickness of delta
y.
So now let us see that where does D go.
D will definitely go to a location D prime
which is what?
Which is somewhat advanced than what?
What was the location of A prime.
So this you can say is like u+delta u*delta
t.
So if you now consider that what is the relative
displacement of D in comparison to A. Why
that is important?
That is important because if you now try to
sketch the new location of AD which is like
this, you realise that it has not only got
displaced linearly.
But it has undergone an angular deformation
which is the so-called shear deformation.
How is this shear deformation quantified?
It is quantified by this angle say delta theta.
If the time interval is small, then obviously
this is expected to be small.
Fluids under shear are continuously deforming.
So here we understand that this kind of deformation
is possible if the fluid is under shear.
So when the fluid is under shear and it is
continuously deforming, you allow more and
more time, this angle will be more and more.
So we restrict to a small time interval when
this was very small and this can be quantified
as tan of delta theta=what?
This neck displacement here is delta u*delta
t/delta y.
For small delta theta, tan of delta theta
is roughly like delta theta and then you divide
that by delta t, right side you have delta
u/delta y.
And you take the limit as delta y tends to
0 as well delta t tends to 0.
So this is nothing but dudy.
If there are other components of velocity,
we are assuming that it is having only a component
of velocity along x, that is not the reality
but to introduce the concept, we have started
with such a simple understanding.
So if it is having only 1 component of velocity,
then this is the case; otherwise, it could
be represented by some partial derivatives.
We will come across the more detailed understanding
of the deformation of the fluid elements when
we will be talking about the kinematics of
fluid flow in a separate chapter but just
for introductory understanding, this in the
right-hand side is a sort of gradient of the
velocity which comes from the velocity profile
and that is representing what?
This is like theta dot.
So it is representing the state of angular
strength of the fluid or rate of angular deformation
so to say.
So when we say rate of deformation of a fluid,
it might be linear deformation or angular
deformation.
If we do not detail it with a further qualification,
we implicitly most of the times mean that
we are talking about angular deformation.
So this is rate of angular deformation of
the fluid.
Now who is responsible for this rate of angular
deformation?
A shear stress, right.
So there is a shear stress.
And the shear stress is very much related
to the disturbance of momentum that was imposed
because of the presence of the plate.
So shear stress may also be interpreted as
a momentum flux.
We will see that how it may be interpreted
with a different example but important thing
to understand is that this theta dot must
be related to the shear stress.
So this is a kind of straining.
So this is rate of shear strain or rate of
angular deformation.
So these are the terminologies which are commonly
used to quantify this one or to exemplify
this one.
So we call this rate of deformation or rate
of angular deformation or rate of shear strain.
In fluids, strain itself is not important
because as we have seen if we allow time,
it will be straining more and more.
So obviously if you want to quantify strain,
it becomes a kind of a redundant exercise.
You allow more time, under shear it will be
straining more and more because fluid is continuously
deforming under shear.
What is most important for a fluid is like
the rate at which it is shearing and for that
this quantification is very very important.
What is responsible for that again, is the
shear stress.
So there must be some relationship between
the shear stress and the rate of deformation.
Why such a relationship is to be present?
Because one is like a cause and another is
like an effect and it is the behaviour of
the material of the fluid that will decide
that how it will respond to a situation and
have an effect of deformation and that type
of behaviour in general continuum mechanics
is known as constitutive behaviour or constitutive
relationship.
That means the fluid has a constitution.
So in a particular disturbance, in a particular
situation, it responds to that and the manner
in which it responds, it comes from its own
constitutive behaviour.
It comes from the material property.
So therefore some relationship which should
relate the rate of deformation with the shear
stress and in a general functional form, we
can write that the shear stress should be
a function of the rate of deformation.
Of course when we are writing a shear stress
here, what should be the correct subscripts
if we want to write it in terms of tau ij
representative.
Say this is x-axis and this is y-axis.
Maybe 12 or 21 because tau 12 and tau 21 are
the same or even you can write tau xy or 12
or 21 or whatever but here we will just omit
the subscript because here we are looking
for only 1 particular component of the stress
tensor.
Other components are not relevant for this.
So we will just call it tau just to be simple
enough in that notation.
Now this function or relationship may be linear,
nonlinear, whatever.
Try to draw an analogy with the mechanics
of solids that you have learnt.
So you have learnt that in most of the solids
which have elastic properties, you have stress
related to strain and that behaviour may be
linear, nonlinear, whatever.
But if you have an elastic material, then
within the proportional limit, you have stress
is proportional to strain and that proportionality
is being connected with an equality through
a material property known as modulus of elasticity.
Of course, all materials are not linear elastic
materials, but this particular law which is
the Hooke's law is a very popular one because
many of the engineering materials will obey
that behaviour within proportional limits.
And many times in engineering, we are walking
within those limits.
Similarly for fluids, very interestingly most
of the engineering fluids that we encounter
and typically the 2 common engineering fluids
we always encounter are air and water and
these fluids will also obey that type of linear
relationship between the shear stress and
the rate of deformation.
So for those fluids which obey the linear
relationship between the shear stress and
the rate of deformation, we call those as
Newtonian fluids.
So Newtonian fluids are those fluids for which
the shear stress is linearly proportional
to the rate of angular deformation or shear
deformation or shear strain.
So for Newtonian fluids, you have tau is proportional
to theta dot.
This proportionality again should be breezed
up with any equality through a material property.
Here the material is a fluid.
So that is expressed by equality through a
fluid property mu which is called the viscosity
of the fluid.
So this is the formal definition of the viscosity
of a fluid.
Of course if it is a Newtonian fluid, then
only this definition works.
So for a fluid which is not a Newtonian one,
then this definition does not work, still
one may cast the relationship in this type
of pseudoform but that is not really a viscosity,
that is called as apparent viscosity because
in a true sense, the viscosity definition
should be following the Newton's law of viscosity
but all fluids do not obey the Newton's law
of viscosity.
The fluids which do not obey the Newton's
law of viscosity are known as non-Newtonian
fluids.
It is an entire branch of science which deals
with how the material should respond in terms
of its shear deformation behaviour and linear
relationship is just only a small part of
that.
The entire science is known as rheology were
you are basically dealing with the constitutive
behaviour of say fluids against various forcing
mechanisms but here we will confine our scope
mostly to Newtonian fluids.
And we will briefly touch upon 1 or 2 examples
of non-Newtonian fluids just to appreciate
that there may be interesting deviations from
the Newtonian behaviour.
To do that, first we will concentrate on this
fluid property viscosity and we will try to
formally find out its units, dimensions and
so on.
So if we try to identify the dimensions of
the viscosity, so it is a dimension of shear
stress, divided by the dimensional of rate
of angular deformation.
So let us write it in the MLT dimension.
So this is stress.
So force per unit area, right, force is mass*acceleration.
So you tell that what should be these.
Whatever you tell, I will write that.
So first the force, MLT-2, then this is what?
This is force divided by area.
So another L to the power -2, then this is
t-1.
So it isů ML-1T-1.
So in SI units, you can write this as kg per
metre second but there are different styles
in which this is written also in SI units.
Of course kg per metre second is one of the
styles but you can also write it in terms
of force units and those are typically more
common units.
So this if you write it in terms of force
units or units of pressure so to say, you
can also write it as Pascal second because
you see this is like that, stress is like
a pressure unit and this is second inverse.
So that goes in the numerator, it may also
be written as Pascal second.
So these are alternative units for the viscosity,
alternative expressions for units of viscosity,
all in SI units.
In our course, we will be always following
SI units and therefore it is important that
we get conversant to these units.
Of course in other units, there are expressions
like in CGS, this particular unit is given
the name as poise and the reason is like these
names have come up to honour the very famous
scientists or mathematicians who developed
the subject of fluid mechanic.
The subject of fluid mechanics has been initially
developed mostly by mathematicians and it
is important to honour them in various ways.
One of the ways is to give units in their
names.
So by giving honour to the Pouseuille, the
famous scientist Pouseuille, the name Poise
came because Pouseuille had many seminal contributions
towards a better understanding of viscosity.
So this is a regarding the units of viscosity
in the SI systems.
Now we will learn a concept which is closely
related to viscosity and that is known as
kinematic viscosity.
That is also a property of the fluid.
So when we say kinematic viscosity, we define
it with a symbol mu which is the viscosity/density,
mu/rho, very simple definition.
We will see that what is the physical significance
of this definition but first let us look into
the units and so on.
So you can see that if you write its SI unit,
what will be its SI unit?
So this was the viscosity unit and rho is
kg per meter cube.
So it will come to SI unit of metre square
per second, right.
It does not have any mass unit involved with
it and that is why the name kinematic because
when you have a mass unit involved, it is
as if like you are kicking of a forcing situation
where the kinetics also come into the picture.
So here it is solely like dictated by units
determining the motion.
So that is why the name kinematics viscosity
but that is something more superficial.
But the concept is more subtle that how we
can utilise the concept of this to get a physical
feel of what is happening within the fluid.
Again take that example where you have like
flow over a plate or any solid boundary.
Now what is the viscosity that tries to do?
It tries to diffuse the disturbance in momentum.
The solid boundary creates a disturbance in
momentum.
Viscosity is a fluid property that tries to
defuse that disturbance in momentum to the
outer fluid.
So that is there in the numerator.
In the denominator, what is there?
In the denominator, there is a fluid property
which tries to maintain its momentum because
it is density.
So it is directly related to the mass and
mass is a measure of inertia.
So what it tries to do?
The denominator represents the physical property
by virtue of which the fluid tries to maintain
its moment and the numerator is a property
by which it tries to disturb its momentum
or propagate the disturbance.
So it is an indicator of the relative tendency
of the fluid to create a disturbance in momentum
as compared to its stability to transmit a
momentum or rather to maintain its momentum,
not to transmit it, to maintain its momentum
through its inertia, okay.
So that is the physical significance of the
kinematic viscosity.
So it is not just solely the viscosity that
is important.
The density is also important because when
you are thinking of the characteristic of
a fluid in transmitting momentum, you must
also try to compare it with a situation where
it is maintaining its own momentum and not
responding to a change in momentum.
So that is why this is a very critical and
important ratio.
The next question that we will ask again qualitatively
that what is the origin of viscosity?
That where from such physical property originates?
Is it just by magic or where from it occurs?
So we will see what is the origin of viscosity
in a very qualitatively way.
Let us first concentrate on liquids.
Why we separately concentrate on liquids and
gases is a very straightforward reason that
the molecule nature of gases and liquids are
different and therefore the viscosity or the
viscous behaviour is going to be different
and it is important to appreciate that at
the end because through a continuum description
like viscosity is a continuum description
of a fluid property.
We are usually abstracted of the molecular
nature but we have to keep in mind that is
the molecular nature of the fluid that has
eventually given rise to this properties.
So there is a direct relationship.
It is like an upscaled version of the molecular
behaviour so that you are abstracted from
it but you have to keep in mind that it is
the molecule behaviour that has given rise
to these properties.
So when we talk about liquids, so when we
talk about liquids, then origin of viscosity
in liquids is intermolecular forces of attraction.
So there are different types of forces of
attraction like if you have like molecules,
like you have unlike molecules, you might
have accordingly different types of forces
like cohesion and adhesion and so on.
Fundamentally if you have 2 different types
of molecules or 2 molecules of the same type,
there will always be some attractive and repulsive
potentials which are acting amongst themselves.
The net effect is an intermolecular force
of attraction that binds the molecular configuration
together; otherwise, molecules will just escape
and for liquids, it is the intermolecular
forces of attraction that gives rise to the
viscosity mainly.
So intermolecular forces of attraction.
However, if we want to give this logic for
gases, it sounds to be weak.
Why it sounds to be weak?
Because we know that gases are much less densely
populated systems and therefore intermolecular
forces of attraction are not that strong.
So what gives rise to a strong viscosity in
gases many times.
Let us look into an example.
Let us say that you have a system of gas molecules.
So one layer of gas molecules which is at
the top of another layer which is like this.
What is the difference between these 2 layers?
The bottom layer is moving at a slower velocity
and the top layer is moving at a faster velocity
and we have seen that how such velocity profiles
occur in a system of fluid.
Now for gases, these molecules have strong
random motion with respect to their mean position.
So these are vibrating with respect to their
mean positions.
So in this way what is happening?
It is very likely that a molecule from the
slower moving layer joins the group of a faster
moving layer and what it will try to do?
It will try to reduce the velocity of the
faster moving layer.
On the other hand, it is very likely that
because of this random thermal motion, this
motion is because of the thermal energy of
the system.
So if the temperature is absolute 0, it will
go to a stop but in any other case, this thermal
motion will be there.
So from the layer, there will be molecules
which are joining the bottom layer and these
will now try to enforce the bottom layer to
move faster.
So that is how there is an exchange or transfer
of molecular momentum and these gives rise
to the viscous properties of gases.
So for gases, it is mostly because of transfer
of molecular momentum.
If we say now that there are certain factors
which are affecting the viscosity for gases
and liquids, then we should be able to explain
how those factors influence the viscosity
through these basic understanding.
Let us concentrate on one of the factors as
an examples.
Let us say temperature.
So viscosity of fluids is generally a strong
function of temperature.
Now let us ask ourselves a question.
If you increase the temperature of a gas and
a liquid, intuitively would you expect the
viscosity to increase or decrease?
Let us take one by one.
Say for liquids, it is expected to decrease.
Why?
If you increase the temperature, the intermolecular
forces of attraction would be overcome by
the thermal agitation and therefore the basic
origin of viscosity will be disturbed.
So it is intuitively expected that for liquids,
the viscosity should decrease with increase
in temperature.
However, for gases what will happen?
If you increase the temperature, there will
be more vigorous exchange of molecular momentum
and therefore for gases, it is expected that
the viscosity will increase with an increase
in temperature.
Of course, there may be exceptions but we
should not discuss exceptions.
These are more rules than exceptions.
So what is the basic understanding.
See whenever we learn a particular concept,
it is important to learn the basic science
that leads to the concept.
So if we just learn it like a magic rule that
viscosity of a liquid decreases with increase
in temperature, it is of no purpose.
Only in your examination, you may answer it
well but the next day you forget and many
times in the examination also, you may confuse
between these 2.
But if you recall what is the correct physical
reasoning that goes behind that, it is very
easy to appreciate that why this should be
more intuitive than not.
Now just as an example for gases, let us try
to see that whether we may quantify the viscosity,
may be for ideal gases, through the concept
of exchange of molecular momentum.
So let us say that you have n as the number
density of the gas molecules.
So let us say n is the number density.
Number density means basically the number
of gas molecules per unit volume.
So with this number density and with these
type of interaction, let us see that what
is the extent of exchange of molecular momentum.
Let us say that m is the mass of each molecule.
Now let us say that these molecules are having
a random thermal motion and the random thermal
motion is being characterised by a velocity
which if you follow the kinetic theory of
gases, it may be expressed as square root
of 8RT/pi for ideal gases.
Now of course, this is a random thermal motion,
so not that this full motion is utilised for
just the transverse velocity component because
the molecules have degrees of freedom, thermal
degrees of freedom in all direction.
So let us say that a fraction of that alpha
is what is utilised for the exchange of transverse
molecular momentum in the y direction.
So if we consider a time of delta t, then
with the time of delta t, what is the distance
that is swept in the vertical direction?
That is alpha*this characteristic velocity*delta
t, that is the distance swept in the y direction
and what is the number of molecules which
is taking part in that interaction.
So this is the distance in the y direction.
If you multiply that with the cross-sectional
area, say delta A of the face that we are
considering across with this molecules are
moving along y, so this represents the total
volume and what is the mass within the total
volume.
So first n is the number density.
So n*this is the number of molecules in the
total volume, that if you multiply with m,
that is the mass of each molecule, then this
is what is the total mass in this volume.
So with this total mass, there is a molecular
momentum of the upper layer.
This *u+delta u of the lower layer is this
*u.
So the net exchange of molecular momentum
is the difference between that 2 and that
is this *delta u.
Now if you want to find out that what is the
rate of exchange of this molecular momentum.
Say per unit time per unit area, that is known
as flux.
So any quantity which you are estimating as
a rate in terms of per unit time and per unit
area normal to the direction of propagation
of that, then that is known as a flux.
So we call it as a momentum flux. so For the
moment flux, this should be divided by delta
A*delta t.
So the momentum flux is nothing but nm alpha
v thermal*delta u.
So this delta u, we may express in terms of
a gradient.
Let us think that we are having interaction
between 2 layer of molecules.
When the interaction is possible, what is
a characteristic length that should separate
them, which would make them interact, that
is the mean free path because that is the
distance over which characteristically one
molecule should traverse before colliding.
So this may be expressed in terms of a gradient
like this.
So this is like the rate of change, that multiplied
by the distance over which it is traversing.
So that is like the characteristic delta u.
So this lambda is the molecule mean free path.
So we have seen that the molecular momentum
flux that is nothing but the shear stress
in a fluid.
So this if it is a Newtonian fluid, this should
also be expressible as equivalent to some
mu*the velocity gradient along y by the Newton's
law of viscosity.
So this is identically equal to shear stress.
Shear stress is nothing but like in this case,
the molecular momentum flux.
So if you equate these 2, then what you get
out of this?
Mu as n*m*alpha*lambda*v thermal.
N*m is what?
It is basically the density, the mass, the
density in terms of mass of the fluid.
So it is rho*alpha*lambda*this thermal velocity.
So it is something which we can just write
in this way.
So this is like roughly an expression that
talks about the viscosity behaviour of ideal
gases and you can clearly see that as the
temperature increases, the viscosity increases.
Of course for real gases, it is not so simple.
For real gases, you also have to consider
other interactions but this just gives a qualitative
picture of the entire scenario.
Keeping this in mind, what we will just do?
We will briefly define a few non-dimensional
numbers with which we will try to correlate
this behaviour.
The first non-dimensional number that we will
be defining is something which you have heard
many times is the Reynolds number.
So when the fluid is flowing, the fluid is
being subjected to different forces.
One of the forcing mechanisms is the inertia
force.
So the fluid has inertia.
Because of that inertia, it tries to maintain
its motion.
On the top of that, there is a resistance
in terms of viscous force which tries to inhibit
the motion.
So we may try to get a qualitative picture
of what is the ratio of this inertia force
and viscous force which will give us an indication
of the extent of the effect of the viscous
force in terms of influencing the fluid motion
when it is subjected to an inertia force.
So the inertia force is like mass*acceleration.
So mass*acceleration, so if you write m*a,
this is the inertia force and the acceleration
in terms of velocity, can be written as like
v square/L, this is also a unit of acceleration.
Just express in terms of velocity.
So mass*acceleration is the inertia force
and the viscous force you can express like
using the Newton's law of viscosity.
So let us try to express the viscous force.
The viscous force is the shear stress*the
area.
Shear stress is mu*the velocity gradient,
so mu*v/L, that is the shear stress, by Newton's
law of viscosity, that *area.
So we are just trying to write it dimensionally.
That into L square is the viscous force.
so this is inertia force and this is viscous
force.
So if you find out the ratio of these 2, what
we will get?
Now when you write the mass*acceleration,
you have to keep in mind that the mass is
the density times the volume.
So the mass is rho*L cube in terms of dimension.
So if you find out the ratio of these 2 forces,
what we will get is rho*V*L/mu.
So this L is what?
It is a characteristic length scale of the
system.
We have earlier discussed that what is the
characteristic length scale of the system.
So in a system with a particular characteristic
length scale, the Reynold's number expressed
in terms of that length scale, and we will
see what is this velocity?
It is also characteristic velocity.
A system may have different velocities at
different points.
So this is a kind of characteristic velocity
taken and with these characterisations, it
may be possible to have a quantification of
the ratio of inertia and viscous force.
Given when the inertia force is absent, this
tells us a way by which we can have the concept
of Reynold's number utilisable that not the
inertia force by viscous force but if it was
possible to have an acceleration by having
a driving force, what would have been that
equivalent inertia force.
The ratio of that equivalent inertia force
by viscous force may be interpreted as a Reynold's
number in cases when the fluid is not accelerating.
So if the entire energy of the fluid was utilised
to accelerate it, what would have been that
equivalent inertia force?
That by viscous force would still then be
interpreted as an equivalent Reynold's number.
So we will try to stop here today and in the
next class, we will try to utilise the concept
of this non-dimensional number and relate
it with the viscous behaviour for gases, okay.
Thank you.
