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So, welcome to this third lecture.
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In the last lecture we looked at Newtonian
fluids, in this lecture we will begin with
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non-Newtonian fluids.
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In the last lecture we introduced Newtonian
fluids as those where the shear stress is
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directly proportional to the rate of deformation,
which was derived and shown to be equal to
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du by dt, du by dy, the velocity gradient.
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Now it is not necessary that for all fluids,
the stress and strain are linearly proportional
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to each other or directly proportional to
each other.
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It may be that du by, the shear stress is
proportional to the rate of deformation which
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is also called the strain rate raised to the
power N.
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So N, Newtonian fluid will be a special case
of this situation where N is equal to 1.
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So, this type of fluids are called non-Newtonian
fluids.
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Now we can write the same equation or same
relation in the form of a Newtonian fluid
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and define a apparent viscosity.
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So, as you see here, we define tau YX, the
shear stress as some constant, this is the
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proportionality constant multiplied by du
by the gradient raised to the power N.
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We can now write it in this form where we
take out modulus of du by dy raised to the
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power N minus1 multiplied by du by dy.
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The idea behind this is to keep this expression
similar to a Newtonian fluid.
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Now, we can write the first part K into du
by dy to the power N minus1 as apparent viscosity.
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So, now if you look at the final equation,
the shear stress is equal to eta into du by
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dy, it looks similar to a Newtonian fluid
but with a difference that now eta is not
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constant, the apparent viscosity is not constant.
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It is dependent on the strain rate du by dy
in some certain way, so it does not remain
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constant, so that is basically a non-Newtonian
fluid.
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The way we have written this equation, you
have, you can see that we have retained the,
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the sign of du by dy, that is also important.
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We have done that by taking modulus of this
quantity du by dy raised to the power N minus1.
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This is also true because the shear stress
should have the same sign or the shear stress
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should be positive if du by dy is positive.
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That means U increase, U increases in the
Y direction, so the velocity is increasing
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in Y direction, it will apply a shear stress
in positive X direction.
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Whereas if the velocity decreases in Y direction
and du by dy is negative, then it will apply
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a shear stress in negative X direction.
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So, that is how the sense of the expression
or the sign of the expression is written by
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using this type of expression, by expressing
it in this way.
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Now let us look at, so we did this so that
we can get an expression similar to the Newtonian
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fluid but with a difference that now we are
talking about apparent viscosity which is
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dependent on the strain rate.
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So, this class of fluids as we have already
introduced are called non-Newtonian fluids.
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Let us look at how the shear stress varies
with the deformation rate in the case of a
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non-Newtonian as well as a Newtonian fluid
and how does the viscosity changes with the
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deformation rate.
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So, if we write, once we have written the
equation in this form, the tau YX is equal
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to eta into du by dy, eta will become the
slope of the variation, slope of this curve
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which plots tau YX with du by dy.
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So, if you consider a Newtonian fluid which
is the simplest of the situation that is N
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is equal to 1, the variation is linear, which
is shown by this Red Line.
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Of course the viscosity remains constant with
the deformation rate, with du by dy, there
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is no variation, there is no dependence on
viscosity on this parameter, it can depend
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on temperature like we saw in the last lecture
but it is independent of the deformation rate,
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so we have plotted viscosity here, the dynamic
viscosity mu which is true for the Newtonian
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fluid or the apparent viscosity eta which
is true for the non-Newtonian fluids.
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Now, depending on the value of N you can have
different types of non-Newtonian fluids.
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So, let us see what are these different types
of non-Newtonian fluids.
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The first possibility is, let us say N is
greater than one.
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So if N is greater than one, what it means
is this expression is increasing with the
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value of the deformation rate.
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So, that means the slope of this curve, shear
stress versus the deformation rate curve will
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increasing, will increase with increase of
the deformation rate, that is apparent from
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this curve itself.
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We can see it more clearly if we plot the
viscosity.
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So, if we plot the viscosity here, the apparent
viscosity of course for the non-Newtonian
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fluid, if we plot it here, we see that for
N greater than one, the viscosity is increasing
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with the deformation rate.
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So, these are called dilatant, non-Newtonian,
type of non-Newtonian fluids.
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So here the viscosity increases with the deformation
rate.
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So, these are called, also called, so by the
application of more shear stress, this liquid
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becomes, this fluid becomes more viscous,
so these are like shear thickening liquids,
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dilatant liquids also called shear thickening
liquids.
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The other possibility is that where N is less
than one, that means you have a situation
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here where the shear stress with deformation
varies nonlinearly like what is shown here
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and the slope of this curve decreases with
increasing deformation rate.
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Okay, if we draw a tangent to this curve,
to this green line at different points, the
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angle of the tangent with the X axis will
reduce, that means the slope of this curve
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as we go for higher deformation rate will
become less.
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In terms of apparent viscosity it means that
the apparent viscosity will reduce, so these
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are called pseudoplastic.
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These are also called, by application of more
shear stress or more deformation rate, this
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becomes thinner.
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So, these are also called shear thinning liquids.
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So, we have seen shear thickening liquids,
types of non-Newtonian fluids and shear thinning
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types of non-Newtonian fluids which are also
known as pseudoplastics.
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There are several examples of this types of
fluids, like for example if you are talking
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about pseudoplastic material, you have paints
for example, when the paint stays in the can,
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it is stationary, it is more viscous.
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When you apply shear, when you apply the paint
on the wall, you are actually applying shear
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on the paint and then under that condition
it becomes thinner, it is also helpful because
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by thinning it becomes easier to apply the
paint on the wall.
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Again there are examples of dilatant fluids
as well, one good example which is familiar
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to us is like, of course cellulose and other
things are also there, one very good example
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is wet sand.
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If you have experienced walking on a wet sand
on the beach, you will see that if you walk
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on the wet sand, your foot will sink.
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But if you jog on the wet sand, the, it becomes,
it is more firm.
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So, when you are walking on the sand what
happens is you are applying a normal force
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and the wet sand cannot, because it is wet,
it gets displaced and the foot sinks.
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But if it is, if you are jogging, you are
applying shear on the wet sand and when you
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apply shear with increasing deformation rate,
it becomes more firm because the viscosity
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is more.
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So, these are some examples of shear thickening
and shear thinning fluids.
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Apart from this there is something called
a Bingham plastic.
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So, what is it, it is actually a plastic,
that means it is like a solid to begin with.
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So what happens, what you can see here is
that, here this curve which shows a Bingham
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plastic actually does not pass through the
origin.
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That means a certain shear stress is required,
a certain value of shear stress is required
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for this material to flow like a fluid.
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It does not flow instantaneously like the
Newtonian fluid, like the dilatant or shear
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thinning or shear thickening, sorry shear
thickening and shear thinning types of fluids.
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So, this needs an initial stress to behave
like a fluid.
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This, the example of Bingham plastic, very
good example is, which we are familiar with
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is toothpaste.
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See, if you, see when it is stored in the
tube, even if you unscrew the lead of the
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tube the toothpaste does not come out or we
do not intend it also to come out automatically.
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So, when it, when you open it, even it does
not come out even if you apply a small force,
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it needs some amount of force for it to come
out.
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So in some sense at that stage it behaves
like a solid, it needs some yield stress for
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it to flow.
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But after it starts flowing, it should not
remain as solid because solids are elastic.
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So what happens?
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If it is, if it behaves like a solid, then
once you put the pressure, it comes out and
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once you withdraw the pressure on the tube,
it goes in because of its elasticity.
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This is not desirable.
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So, under that condition it flows, it flows
like a Bingham plastic, it undergoes permanent
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deformation, it comes out.
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So, this is an example of a Bingham plastic.
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Mathematically if you want to represent it,
it is little, of course it will be little
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different from this expression, so it is something
like this.
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The shear stress, you have some constants,
is now, there is a constant so that means
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even if the deformation rate is zero, you
need a kind of yield stress for this to start
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flowing.
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So this is basically a mathematical expression
for a Bingham plastic.
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So, the situation, so now we see that the
way shear stress and strain rate relates to
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each other in case of fluid is quite complicated,
it is not restricted to just Newtonian behaviour.
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Of course in this course we will be restricting
to only Newtonian fluids but we should know
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that it can behave in various situations in
a non-Newtonian way.
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So far whatever we have discussed brings out
the dependence of the apparent viscosity on
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the strain rate or the rate of deformation.
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But, if you see, the statement here, it says
the apparent viscosity eta could be time-dependent,
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that means even if the shear stress applied,
applied shear stress of the rate of deformation
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is same, it can change with time.
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So, the examples, so this time dependency
brings in more complicacy in the type of behaviour
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for the non-Newtonian fluids.
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The first type, of course there are two possibilities
when we are talking about the time dependency
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of apparent viscosity, it can decrease with
time, these are known as Thixotropic, many
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paints for example are thixotropic which helps
because if you keep on applying even at a
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same shear rate, it becomes thinner.
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And it could be Rheopectic, that means eta
increases with time.
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Apparent viscosity increases with time, this
is also possible.
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So, in this particular slide we looked at
different types of behaviour of stress with
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deformation rate.
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We looked at the stress field before in the
last lecture and we also looked at the stress
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strain relationships, stress and strain rate
relationship for a Newtonian fluid to in,
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today we have introduced non-Newtonian fluids
also.
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The next important thing for this first part
of this module is how do you classify the
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flows.
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We have introduced to what is fluid and what
are the different flow fields, what are the
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different types of velo parameters characterizing
the flow field like the velocity field, stress
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field, etc., how the stress field relates
to the velocity field.
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Now there are different ways of classifying
flows.
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Let us look at some of these classifications.
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The first classification is very much related
to our last discussion, we introduced this
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property called viscosity.
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Now we can talk about viscous versus inviscid
flows.
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So, we can say the viscous flow and we can
say, we can consider inviscid flows also.
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Of course in reality no flow or no fluid is
completely inviscid.
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It is understandable.
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But then what do we mean by inviscid flow
and why do we want to study inviscid flow?
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To look into that, let us look at this example
of an aerofoil.
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This is a particular flat bottom aerofoil.
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Now if you consider a flow past this aerofoil.
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You can consider this as hydrofoil also, that
means water flowing across this particular
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object or air flowing across this particular
object which is an aerofoil.
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So, now if we consider a flow, we can define
a region which is shown by this dashed line
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and this region is the region where the viscous
behaviour is only important.
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That means the fluid behaves in a viscous
way.
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What we mean by that?
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So, it means that the viscous stresses are
important only in this region which is a thin
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region around the aerofoil.
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What happens outside?
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There is still a flow outside.
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Okay.
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So, this region where the viscous forces are
important or viscosity is important is known
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as the boundary layer.
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We will talk about boundary layer later in
this first module of this course but for the
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time being we can just take it as a region
where the viscosity plays an important role
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within a flow.
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Now, outside boundary layer what happens?
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Outside the boundary layer on the flow is
inviscid.
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So, this brings in the importance of viscous
and inviscid flow.
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Essentially an inviscid flow does not mean
that the viscosity of the fluid is zero, it
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means that the viscous forces are not so important
to consider there.
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For example, if this is a aerofoil and air
is flowing across past this aerofoil, the
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viscosity is uniform, with, is same within
and outside the boundary layer, within the
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boundary layer or outside the boundary layer.
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The same viscosity of, or if it is a hydrofoil,
then, let us say water is flowing across this
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hydrofoil, the same viscosity is there for
the fluid water inside and outside the boundary
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layer.
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But what is different is the behaviour in
terms of viscous forces.
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We can neglect the viscous stresses or viscous
forces in this region, in the inviscid flow,
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in the inviscid flow region.
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But within the viscous boundary within the
boundary layer, viscous forces are dominant
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and they have to be considered.
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So, this is the first classification and,
yah, this inviscid flow is very important
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to analyse because often we can see particularly
in Aerodynamic application, we see that you
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can actually explain the lift just by considering
the flow to be inviscid.
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So, what decides, majorly decides lift is
the flow outside the boundary layer, not inside
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the boundary layer.
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So, that is why inviscid flow has its own
importance of study.
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The next classification is based on the fact
whether the flow, whether it is a compressible
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or an incompressible flow.
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Now, we have, we have introduced compressibility
of the fluid in terms of bulk modulus in our
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first lecture.
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00:20:04,260 --> 00:20:12,820
Let us extend that discussion to make this
distinction between the compressible and incompressible
215
00:20:12,820 --> 00:20:13,850
flow.
216
00:20:13,850 --> 00:20:21,679
So, this compressibility of the fluid, it
was defined before as the reciprocal of bulk
217
00:20:21,679 --> 00:20:22,679
modulus.
218
00:20:22,679 --> 00:20:32,960
K is the bulk modulus and if you try to write
it now, it means that this is given as the
219
00:20:32,960 --> 00:20:37,670
relative change in density divided by the
applied pressure.
220
00:20:37,670 --> 00:20:47,399
Now, if you look at this expression for compressibility,
we had also introduced in during the previous
221
00:20:47,399 --> 00:20:54,190
lecture that if you take an example of air
as a fluid, it has very low value of bulk
222
00:20:54,190 --> 00:21:00,270
modulus and it has, that is why it is highly
compressible, whereas water has higher value
223
00:21:00,270 --> 00:21:06,929
of bulk modulus, it has, it is less compressible,
compressibility is less.
224
00:21:06,929 --> 00:21:13,960
But that does not mean that whenever we consider
air, we have to consider the flow to be compressible,
225
00:21:13,960 --> 00:21:19,240
the flow can still be incompressible but the
fluid can be compressible.
226
00:21:19,240 --> 00:21:23,580
Now what is meant by this?
227
00:21:23,580 --> 00:21:30,650
This actually means, this actually brings
us to the definition of a compressible flow
228
00:21:30,650 --> 00:21:32,470
or an incompressible flow.
229
00:21:32,470 --> 00:21:40,590
So, an incompressible flow means that the
flow induced density variation is small.
230
00:21:40,590 --> 00:21:48,070
So, there is density variation in the flow,
let us say the flow is that of air, it is
231
00:21:48,070 --> 00:21:55,510
highly compressible fluid but even that fluid,
the flow of that fluid could be incompressible
232
00:21:55,510 --> 00:21:59,720
if the flow induced density variation is small.
233
00:21:59,720 --> 00:22:06,269
Now, what do we mean by flow induced density
variation?
234
00:22:06,269 --> 00:22:13,400
What is meant by, meant by that is that we
are talking of, let us look at this definition,
235
00:22:13,400 --> 00:22:21,230
so we are talking of this pressure if this
change in pressure is brought about by the
236
00:22:21,230 --> 00:22:30,769
flow itself and that variation is less, then
you can consider the flow to be incompressible.
237
00:22:30,769 --> 00:22:35,510
So, let us do it little bit more mathematically.
238
00:22:35,510 --> 00:22:42,419
From this definition of compressibility of
the fluid, we can write this expression, we
239
00:22:42,419 --> 00:22:50,510
can write delta rho by rho is del P, delta
P by K, directly from this expression we can
240
00:22:50,510 --> 00:22:51,510
write.
241
00:22:51,510 --> 00:22:57,649
So that means any change in density is due
to a change in pressure.
242
00:22:57,649 --> 00:23:04,400
and this change in density and pressure are
related through this expression.
243
00:23:04,400 --> 00:23:13,590
Now, bulk modulus can be also related to the
velocity of sound through that fluid.
244
00:23:13,590 --> 00:23:21,020
So, velocity of sound through a particular
fluid is given by square root of K by rho,
245
00:23:21,020 --> 00:23:27,970
that means K, that is bulk modulus can written
as rho into C square.
246
00:23:27,970 --> 00:23:33,450
Of course we saw bulk modulus has a unit of,
we can check the consistency of units here
247
00:23:33,450 --> 00:23:41,080
also, K has a unit of pressure and rho into
velocity square, also you can check it yourself
248
00:23:41,080 --> 00:23:44,940
as a unit of pressure.
249
00:23:44,940 --> 00:23:52,509
Okay, so K is rho into C square, C is the
velocity of sound through the medium, through
250
00:23:52,509 --> 00:23:57,940
the fluid which we are talking about, for
which we are ascertaining the compressible,
251
00:23:57,940 --> 00:24:02,289
whether it is a compressible or an incompressible
flow.
252
00:24:02,289 --> 00:24:12,519
Now, we will introduce this later but actually
you can write delta P, if it is, that means
253
00:24:12,519 --> 00:24:18,679
the flow induced delta P, pressure variation
can be written, can be scaled with rho V square
254
00:24:18,679 --> 00:24:19,809
by 2.
255
00:24:19,809 --> 00:24:20,809
Okay.
256
00:24:20,809 --> 00:24:28,750
So, in the next chapter we will actually next
to next chapter we will be introducing this
257
00:24:28,750 --> 00:24:29,750
relation.
258
00:24:29,750 --> 00:24:32,700
So, delta P can be related to rho V square
by 2.
259
00:24:32,700 --> 00:24:39,440
So, if the velocity in the fluid is V, then
the pressure variation can be estimated from
260
00:24:39,440 --> 00:24:41,190
there using this relation.
261
00:24:41,190 --> 00:24:45,000
The pressure variation will be rho V square
by 2.
262
00:24:45,000 --> 00:24:50,029
If you plug in these two expressions in this
equation, what you get is like this.
263
00:24:50,029 --> 00:24:57,320
You get delta rho by rho is basically, so
you plug in delta rho, delta P as rho V square
264
00:24:57,320 --> 00:25:05,640
by 2 and K as rho C square, rho cancels out
and we get half V by C whole square.
265
00:25:05,640 --> 00:25:09,340
V by C is defined as Mach number.
266
00:25:09,340 --> 00:25:16,809
So, this is basically the velocity of the,
velocity in the fluid divided by velocity
267
00:25:16,809 --> 00:25:19,549
of sound through the fluid.
268
00:25:19,549 --> 00:25:25,529
Now this can be, dell rho by rho can written
as then M square by 2, Mach number square
269
00:25:25,529 --> 00:25:27,130
by 2.
270
00:25:27,130 --> 00:25:37,290
Now, let us say dell rho by rho, this is just
an assumption that if we see that we can call
271
00:25:37,290 --> 00:25:44,960
the flow as incompressible if the percentage
change in density relative change in density
272
00:25:44,960 --> 00:25:48,240
is more than 5%, less than 5%.
273
00:25:48,240 --> 00:25:53,419
We can call it compressible if it is more
than 5%, the relative change in density is
274
00:25:53,419 --> 00:25:55,139
more than 5%.
275
00:25:55,139 --> 00:26:02,600
So, here as we are trying to find out the
criteria for incompressible flow, we can see
276
00:26:02,600 --> 00:26:07,889
delta rho by rho should be less than 5% or
0.05.
277
00:26:07,889 --> 00:26:08,990
Okay?
278
00:26:08,990 --> 00:26:14,549
So, 0.05 for the flow to be incompressible.
279
00:26:14,549 --> 00:26:20,039
Now if we use this expression here, we can
get a value of Mach number.
280
00:26:20,039 --> 00:26:23,799
You can directly plug in here, you can get
a value of Mach number.
281
00:26:23,799 --> 00:26:28,100
Should be less than 0.3 for an incompressible
flow.
282
00:26:28,100 --> 00:26:37,750
So, even if the flow is, even if the fluid
is highly compressible like air, the flow
283
00:26:37,750 --> 00:26:41,590
could be incompressible if the Mach number
is less than 0.3.
284
00:26:41,590 --> 00:26:54,100
And the reason for this, this 0.3 value is
a well accepted value for 5% change of density,
285
00:26:54,100 --> 00:26:58,880
relative change of density.
286
00:26:58,880 --> 00:27:06,070
This is a well accepted value in different,
in the academic, by the fluid dynamists.
287
00:27:06,070 --> 00:27:12,460
So, this is, this can be applied as a criteria
and you can find out if you know the velocity,
288
00:27:12,460 --> 00:27:19,300
say the maximum velocity in the flow and you
know the velocity of sound through that medium,
289
00:27:19,300 --> 00:27:24,149
you can find out what is the Mach number and
say that if it is less than 0.3, you can simply
290
00:27:24,149 --> 00:27:29,129
use incompressible flow assumption even for
a compressible fluid.
291
00:27:29,129 --> 00:27:38,340
So, this is the second type of classification
for flow, whether it is compressible or incompressible.
292
00:27:38,340 --> 00:27:40,549
There are further flow classifications.
293
00:27:40,549 --> 00:27:44,379
One is laminar versus turbulent flow.
294
00:27:44,379 --> 00:27:48,710
So, let us see what is laminar flow and what
is turbulent flow.
295
00:27:48,710 --> 00:27:51,370
We just introduced these concepts here.
296
00:27:51,370 --> 00:27:59,230
So, a laminar flow means the fluid particles
move in smooth layer, they are, they move
297
00:27:59,230 --> 00:28:00,309
in laminas.
298
00:28:00,309 --> 00:28:02,009
How do you know that it moves in lamina?
299
00:28:02,009 --> 00:28:04,960
We will show that quickly.
300
00:28:04,960 --> 00:28:11,090
But before that the turbulent flow is one
in which the fluid particles move randomly
301
00:28:11,090 --> 00:28:17,460
because there is a lot of fluctuation, velocity
fluctuations, three-dimensional velocity fluctuations
302
00:28:17,460 --> 00:28:19,289
in the flow.
303
00:28:19,289 --> 00:28:27,509
So, now let us see how you can visualize this.
304
00:28:27,509 --> 00:28:34,230
So, if you consider a path line of a particle,
we already defined a path line, a path line
305
00:28:34,230 --> 00:28:41,669
is basically the line traced by a particle
introduced at a particular point in the flow.
306
00:28:41,669 --> 00:28:47,450
So, if you draw a path line of a particle
in a laminar flow, if the flow is straight
307
00:28:47,450 --> 00:28:53,799
like this, it is smooth like this, it need
not be straight, it can be curved also but
308
00:28:53,799 --> 00:28:55,990
it is smooth.
309
00:28:55,990 --> 00:28:58,010
What do you mean by this smoothness?
310
00:28:58,010 --> 00:29:02,840
We can easily see it if we see the path line
for the case of a turbulent flow.
311
00:29:02,840 --> 00:29:07,879
So, for a turbulent flow, for a similar type
of turbulent flow, let us say flow through
312
00:29:07,879 --> 00:29:11,150
a pipe, the pipeline will be something like
this.
313
00:29:11,150 --> 00:29:20,000
It will be very rough, it will move through,
it will move haphazardly through the flow.
314
00:29:20,000 --> 00:29:27,320
Of course there are the experiments for this
were done by, the pipe flow experiment, you
315
00:29:27,320 --> 00:29:35,380
may be familiar with, were done by Osborne
Reynolds and those experiments according to
316
00:29:35,380 --> 00:29:44,140
the name of Reynolds in like we defined Mach
number four compressible flow and incompressible
317
00:29:44,140 --> 00:29:48,500
flow, we define Reynolds number to ascertain
whether the flow is laminar or turbulent.
318
00:29:48,500 --> 00:29:55,100
So, the Reynolds number is defined as inertia
force by viscous force.
319
00:29:55,100 --> 00:30:03,600
You can show that it becomes, this ratio of
the forces comes out to be mathematically
320
00:30:03,600 --> 00:30:10,500
in this way rho, density, velocity, length
scale, so for a pipe flow, the length scale
321
00:30:10,500 --> 00:30:12,350
will be the diameter of the pipe.
322
00:30:12,350 --> 00:30:19,649
For a flow over a flat plate, it is the length
of the plate divided by dynamic viscosity.
323
00:30:19,649 --> 00:30:25,059
Of course this is a non-dimensional number
and the value of this number will say whether
324
00:30:25,059 --> 00:30:31,409
the flow is laminar or turbulent.
325
00:30:31,409 --> 00:30:40,600
So
for example in the case of a flow through
326
00:30:40,600 --> 00:30:49,070
a smooth pipe, we can distinguish between
the laminar and turbulent flow in this manner.
327
00:30:49,070 --> 00:30:55,830
This region, that means less than Reynolds
number, less than 2300 is known as a laminar
328
00:30:55,830 --> 00:31:03,610
flow, is a laminar flow, of course if you
have a rough pipe, these numbers may change.
329
00:31:03,610 --> 00:31:09,370
And it is not like that if it is more than
2300, immediately the flow becomes fully turbulent,
330
00:31:09,370 --> 00:31:14,799
It goes through a transition region, roughly
at a value of 4000 more than 4000, it becomes,
331
00:31:14,799 --> 00:31:23,149
the flow becomes fully turbulent for the case
of a flow through a smooth pipe.
332
00:31:23,149 --> 00:31:29,039
One thing I forgot is that what is the rationale
behind the definition of this number inertia
333
00:31:29,039 --> 00:31:30,700
force by viscous force.
334
00:31:30,700 --> 00:31:36,519
How is that able to determine whether the
flow is laminar or turbulent?
335
00:31:36,519 --> 00:31:43,129
So, a very simple answer to this question
is that the inertia force is something which
336
00:31:43,129 --> 00:31:49,340
tries to make the flow haphazard or random,
like what is shown in the path line of turbulent
337
00:31:49,340 --> 00:31:50,720
flow.
338
00:31:50,720 --> 00:31:58,259
Viscous force, viscous is a damping as we
have seen while defining a fluid, so because
339
00:31:58,259 --> 00:32:04,529
the damping nature of the viscosity, it tries
to reduce this kind of fluctuation.
340
00:32:04,529 --> 00:32:12,120
So, a relative value of inertia and viscous
tells us whether the flow is laminar or turbulent.
341
00:32:12,120 --> 00:32:21,070
If the inertia is more, then it becomes, it
tends to become more turbulent, if the viscous
342
00:32:21,070 --> 00:32:25,610
force is more, then it moves towards a laminar
flow.
343
00:32:25,610 --> 00:32:36,460
That is a very simple explanation for the
fact how Reynolds number can be used to classify
344
00:32:36,460 --> 00:32:40,889
laminar and turbulent flows.
345
00:32:40,889 --> 00:32:46,490
Apart from this there are further ways you
can classify a flow.
346
00:32:46,490 --> 00:32:53,279
One thing is quite useful in fluid mechanics
is internal versus external flows.
347
00:32:53,279 --> 00:32:56,559
So, what we mean by internal flows?
348
00:32:56,559 --> 00:33:01,559
Internal flows are basically wall bounded
flows, that means the flow is surrounded by
349
00:33:01,559 --> 00:33:03,850
a wall.
350
00:33:03,850 --> 00:33:06,789
External flow are unbounded flow.
351
00:33:06,789 --> 00:33:10,539
So, why do we need to classify these two flows?
352
00:33:10,539 --> 00:33:13,259
Before going to that let us take an example.
353
00:33:13,259 --> 00:33:20,340
The example is again this pipe flow, so the
flow inside the pipe is the actually an internal
354
00:33:20,340 --> 00:33:21,370
flow.
355
00:33:21,370 --> 00:33:28,179
When it comes out, it forms a jet here, this
jet is an example of a external flow.
356
00:33:28,179 --> 00:33:34,470
Inside the pipe, it is bounded by the walls
of the pipe, outside the pipe, it is unbounded,
357
00:33:34,470 --> 00:33:38,220
so this flow takes place in this, it extends
everywhere.
358
00:33:38,220 --> 00:33:43,769
So, there is no boundary, so this is called
an external flow.
359
00:33:43,769 --> 00:33:50,100
Now, why, so this is internal flow and this
is external flow.
360
00:33:50,100 --> 00:33:55,009
Now, why do you need to classify these two
flows?
361
00:33:55,009 --> 00:34:04,930
I can explain that using what we have just
now talked about, we talked about laminar
362
00:34:04,930 --> 00:34:13,570
versus turbulent flow, so if we take an example
of an internal flow through a pipe, as we
363
00:34:13,570 --> 00:34:17,599
saw here, the laminar for a, the flow to be
laminar, the Reynolds number should be less
364
00:34:17,599 --> 00:34:21,710
than 2300.
365
00:34:21,710 --> 00:34:27,990
Let us consider a flow over a flat plate,
for this flow to be laminar, so this example
366
00:34:27,990 --> 00:34:35,089
of flow over a flat plate is again an unbounded
flow because although you have a plate on
367
00:34:35,089 --> 00:34:38,669
one side, the other side is unbounded.
368
00:34:38,669 --> 00:34:43,419
As you go perpendicular to the plate, there
is no boundary.
369
00:34:43,419 --> 00:34:47,919
If you put a plate on the top, it becomes
a flow through a channel but if you do not
370
00:34:47,919 --> 00:34:52,460
put, it is just a flow over a flat plate,
it is an unbounded flow.
371
00:34:52,460 --> 00:34:59,200
So, if you take an example of flow over a
flat plate, you can see for the flow to be
372
00:34:59,200 --> 00:35:04,550
laminar, the Reynolds number should be less
than 3 into 10 to the power 5.
373
00:35:04,550 --> 00:35:08,690
Of course Reynolds number is defined using
different length scale, for, which is the
374
00:35:08,690 --> 00:35:13,829
length of the plate in the case of a flow
over a flat plate.
375
00:35:13,829 --> 00:35:21,109
But you can see that the, these two flows
are, flows have very different characteristics.
376
00:35:21,109 --> 00:35:25,130
That is why we need to study this flow in
a different way.
377
00:35:25,130 --> 00:35:30,960
The number designating the transition from
laminar to turbulent, that is Reynolds number
378
00:35:30,960 --> 00:35:37,250
in these two case have very different values
for an internal flow and an external flow
379
00:35:37,250 --> 00:35:38,250
like this.
380
00:35:38,250 --> 00:35:47,180
So, we need to distinguish and study them
separately.
381
00:35:47,180 --> 00:35:57,260
Before ending this first part we also want
to use the concept of the integral and differential
382
00:35:57,260 --> 00:36:04,130
analysis because this is what we are going
to do in the next two chapters of our study
383
00:36:04,130 --> 00:36:07,589
of fluid dynamics.
384
00:36:07,589 --> 00:36:13,710
So, there are, these are two main ways of
analyzing fluid flows.
385
00:36:13,710 --> 00:36:18,059
So, what do you mean by this integral and
differential analysis?
386
00:36:18,059 --> 00:36:24,910
So, we just list the characteristics of integral
analysis here and differential analysis on
387
00:36:24,910 --> 00:36:30,050
the right hand side.
388
00:36:30,050 --> 00:36:35,290
In the case of, main difference is that, in
the case of an integral analysis, the basic
389
00:36:35,290 --> 00:36:42,770
laws, conservation laws which we will introduce
very soon are applied to finite size control
390
00:36:42,770 --> 00:36:43,770
volume.
391
00:36:43,770 --> 00:36:50,160
We will give example of this finite size but
in the case of a differential analysis, these
392
00:36:50,160 --> 00:36:59,360
basic laws are applied to infinitesimal, infinitesimal
control volumes, very small control volumes.
393
00:36:59,360 --> 00:37:07,160
So and what happens due to this different
consideration of the size of the control volume?
394
00:37:07,160 --> 00:37:14,020
What happens is the output here, we try to
do this analysis where we intend to get output
395
00:37:14,020 --> 00:37:16,070
like overall quantities.
396
00:37:16,070 --> 00:37:24,220
Like the force, force in terms of drag on
a plate or drag on a aeroplane or lift on
397
00:37:24,220 --> 00:37:27,890
a hydrofoil, whatever it is, or torque, things
like this.
398
00:37:27,890 --> 00:37:33,290
So, these overall quantities are the parameters
of interest.
399
00:37:33,290 --> 00:37:38,550
Whereas in the case of the differential analysis,
the parameters of interest are different because
400
00:37:38,550 --> 00:37:44,790
we have taken very small control volumes,
we can get flow field.
401
00:37:44,790 --> 00:37:51,050
So we can get velocity field, we can get pressure
field and so on and so forth, temperature
402
00:37:51,050 --> 00:37:53,069
field etc.
403
00:37:53,069 --> 00:37:58,890
So, the depending on what we want to get from
the analysis, if we are interested in these
404
00:37:58,890 --> 00:38:08,050
overall quantities, we go for an integral
analysis, if you want to get detailed information
405
00:38:08,050 --> 00:38:16,250
like the field information, flow field information,
we go for a differential analysis.
406
00:38:16,250 --> 00:38:22,859
We take an example of this, some examples
of where we can apply integral analysis.
407
00:38:22,859 --> 00:38:28,960
So, this is a Pelton turbine and then we can
see, if you take, so this is the direction
408
00:38:28,960 --> 00:38:36,010
of the flow, it is hitting this turbine blade
and the turbine is rotating.
409
00:38:36,010 --> 00:38:42,460
Now if you consider just one of this, you
can consider a control volume across this
410
00:38:42,460 --> 00:38:46,500
and you can find out force on the turbine
wheel, this is very useful because this will
411
00:38:46,500 --> 00:38:54,140
help us to find out from the, how the torque
generated in the turbine is related to the
412
00:38:54,140 --> 00:39:02,520
velocity of the flow hitting the turbine blades,
by using a control volume analysis.
413
00:39:02,520 --> 00:39:07,329
So, the control volume, actually we will introduce
in the beginning of the next chapter but it
414
00:39:07,329 --> 00:39:15,470
essentially means this is a fixed region in
space which can exchange both mass energy
415
00:39:15,470 --> 00:39:18,250
with the rest of the space.
416
00:39:18,250 --> 00:39:22,830
So, by doing this analysis we can find out
force like this.
417
00:39:22,830 --> 00:39:30,920
For this case, or let us say we have a rocket
and we can take a control volume like this
418
00:39:30,920 --> 00:39:37,809
in the rocket and find out the thrust acting
on the rocket.
419
00:39:37,809 --> 00:39:44,520
Similarly we can take an aeroplane and we
can take the wings of the aeroplane, the aerofoil
420
00:39:44,520 --> 00:39:50,319
section, the flow across the aerofoil section
and we can find out the lift on the aeroplane
421
00:39:50,319 --> 00:39:51,319
wing.
422
00:39:51,319 --> 00:39:58,369
So, this kind of information definitely are
very useful to us, these are just three examples
423
00:39:58,369 --> 00:40:04,810
and for you can go on like this and then these
are very useful to us.
424
00:40:04,810 --> 00:40:10,880
On the other hand, this is also useful, differential
analysis is also very useful when we want
425
00:40:10,880 --> 00:40:14,650
to look at the details of the flow, like say
the flow field.
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00:40:14,650 --> 00:40:21,290
This is an example, like flow over a flat
plate, we want to see how the velocity varies
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00:40:21,290 --> 00:40:27,020
along the Y direction, this is the velocity
profile like we demonstrated this before while
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00:40:27,020 --> 00:40:28,730
introducing a timeline.
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00:40:28,730 --> 00:40:35,710
So, this is like how the velocity varies,
if you go in the direction perpendicular to
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00:40:35,710 --> 00:40:37,000
this wall.
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00:40:37,000 --> 00:40:41,859
So, when we want this detailed analysis and
how does it vary along X direction also.
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00:40:41,859 --> 00:40:45,730
So, when we want the detailed velocity field,
we do a differential analysis.
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00:40:45,730 --> 00:40:53,270
So, this brings us to the end of the third
lecture.
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00:40:53,270 --> 00:41:00,310
In this lecture we have looked at, we have
just started with what we ended with in the
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00:41:00,310 --> 00:41:05,140
last lecture where we introduced Newtonian
fluids.
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00:41:05,140 --> 00:41:13,040
In the in this lecture we started with non-Newtonian
fluids, different types of non-Newtonian fluids,
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00:41:13,040 --> 00:41:18,450
their behaviour, how the stress and the rate
of the deformation are related for different
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00:41:18,450 --> 00:41:25,510
types of non-Newtonian fluids and we looked
at flow classifications like compressible
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00:41:25,510 --> 00:41:33,721
versus incompressible flow, viscous versus
inviscid flow, then so and so forth, internal
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00:41:33,721 --> 00:41:40,670
versus external flow, laminar versus turbulent
flow, these 4 classification we looked at.
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00:41:40,670 --> 00:41:45,310
These are the major ways of classifying fluid
flows.
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00:41:45,310 --> 00:41:52,260
There could be more ways also but these are
the main ways generally we classify fluid
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00:41:52,260 --> 00:41:53,260
flows.
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00:41:53,260 --> 00:41:59,400
And we have also looked at 2 different types
of analysis and this is very pertinent to
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00:41:59,400 --> 00:42:06,510
us in terms of what we are going to talk about
in the next 2 chapters.
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00:42:06,510 --> 00:42:12,150
In the next chapter we take up integral analysis
and then we will take up differential analysis
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00:42:12,150 --> 00:42:13,220
in the 3rd chapter.
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00:42:13,220 --> 00:42:14,069
Thank you.