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Welcome to this lecture number 8 on this NPTEL
course on fluid mechanics. For fluid mechanics;
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for under graduates students in chemical engineering
and until the last lecture we discussed fluids
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under static conditions, the variation of
pressure in a fluid under static conditions,
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the forces that are experienced by objects
submerged in a fluid under static conditions
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and so on. Before, I move on to the new topic,
I am going to quickly recapitulate the key
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results that we discussed so far.
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So, under fluid statics, we first saw that
if a fluid is under static conditions, you
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should take any volume element. This is a
volume element in a fluid. Then, the forces
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that are exerted by the fluid that is present
outside on the fluid that is present inside
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is purely normal and it is acting in a compressive
sense. So, the force that is exerted for the
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fluid that is present outside on the fluid
is present inside is denoted by R. And if
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you take any particular point on this volume
element, this force is a function of the unit
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normal. So, since the; so, if you take any
point on this surface, the unit normal point
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outwards that it is from inside to outside.
The force on a static fluid element acts from
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outside to inside.
So, this force is written is written as p
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times minus n because minus n is the direction
at which the force is acting and p is the
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magnitude of the force. Well, it is more appropriate
and this is the magnitude of compressive force
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per unit area acted upon by the fluid outside
on the fluid that is inside, on the surface
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that separates outside and inside. So, this
compressive force per unit area, the magnitude
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of it is called the pressure in a fluid. And
we saw that this pressure has units, pascals.
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So, 1 Pascal is 1 Newton per meter square.
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And then we said that if you have a fluid
that is under the influence of a gravitational
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field. So, we took a coordinate system x,
y and z and gravity acts in the direction
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of minus z direction. So, if you look at;
if you want to write the gravity vector. It
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is minus g times k. The g is the acceleration
due to gravity, it is 9.8 meter per second
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squared and the surface of the earth and k
is the direction of the positive z axis. Minus
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k is the direction of negative z axis. So,
g is minus, the g vector is minus g times
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k. So, we have a fluid that is present under
the influence of a gravitational field and
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we said that by taking tiny volume element
of a fluid. We can show that the fundamental
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equation of hydrostatics is minus del p plus
rho g is equal to 0.
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So, this is the fundamental equation for a
fluid under static conditions. This the physical
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interpretation for this equation is as it
is very simple rho g is the force per unit
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volume, volume due to gravity. This is the
weight of the tiny volume element and this
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is del p is the net pressure for force minus
del p. So, net pressure force per unit volume
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acting on the fluid element and since a fluid
element is at rest the sum of all the forces
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must be 0 and that is what gives rise to this
simple balance.
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Now, normally this equation, if you remember
that this is very general equation in the
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sense that this is valid for any coordinate
system. Not necessarily the rectangular coordinate
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system depicted in this cartoon valid for
any coordinate system. So, but if you want
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to solve a problem you refer this equation
with respect to this partition or a rectangular
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coordinates x y z and g pointing like this.
So, we found that minus dp dz minus rho g
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is 0 or dp dz is minus rho g.
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So, if you want to integrate this equation
we have to assume certain things about density.
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If you assume incompressible flow, sorry,
incompressible fluids for which density is
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independent of pressure, rho is independent
of pressure p. Then we can and since g is
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a constant both rho and g are constants in
this equation. So, we can easily integrate
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this. So, you get integral dp between any
2 points p1 or p0 and p is minus rho g integral
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z naught to z dz. So, p minus p0 is rho g
times z0 minus z. Now, in most applications
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you have a pool of liquid that is water let
say and that is open to an atmosphere, air
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at atmospheric pressure.
So, this is p atmosphere, this is the free
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surface that separates the liquid from air,
water from air. For example, so at this point
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the pressure is p atmosphere, it is known.
So, our coordinate system is align like this,
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x y z, where z is increasing in this direction.
So, if you refer z naught in this equation
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as the free surface z equals to z naught and
any point z, is this distance is z naught
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minus z and we can call that as h.
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h is the depth of the liquid from the
free surface. In which case, we can rewrite
this equation as p minus p0 atmosphere because
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that is the value the pressure at z equals
to z0. p minus p atmosphere is rho gh or p
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is p atmosphere plus rho gh. This is a fundamental
equation valid for incompressible fluids because
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we assume rho is constant and it is valid
when gravity is in the direction of minus
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k. So, these are the two key assumptions and
then this equation is very useful in solving
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many problems.
One example is what we did was to show, suppose
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you have a free surface and then you have
a solid surface, solid planar solid surface.
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And then we are interested in finding the
force, net force, effective force on this
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solid surface and that force will act normally
what is the magnitude of force which we call
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FR and what is the line of action of the force?
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So, both of this we calculated by simply.
So, this is a cross section of a planar surface
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like this. So, this is the planar surface
and you take a tiny area element and then
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we find what is the; that is called as dA.
The pressure the force acting is p dA and
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then you integrate this you get the total
force FR. So, it is as simple as said and
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then by equating the moments we found what
is the line of action the last lecture. The
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next application of this simple formula that
p is p atmosphere plus rho gh was to calculate
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the force on a submerged object.
Suppose, you have a solid object that is submerged
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in a fluid and this is atmospheric pressure.
The fact is that because gravity is acting
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in this direction, the force the pressure
on this side will be smaller than the pressure
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on this side because of this column of liquid
that increases the pressure by rho gh.
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Then, we saw that by taking a tiny cylindrical
volume element and we saw that the net force
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acts upwards because of the fact that the
pressure is more than that the pressure here.
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And by taking many such tiny cylindrical volumes
over the entire volume, we saw that the net
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force acting upwards, the direction upwards
to the gravity vector. That is the upward
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direction upwards on a submerged object is
simply rho of the liquid, density of the liquid
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times g times the volume of the liquid that
is been displaced by the solid. Here the entire
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volume is submerged. So, the entire volume
of the solid itself volume of the liquid that
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is displaced by the solid. If the object is
partially submerged like here only this part
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is submerged, then is a floating object.
By the by the way this force is called the
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buoyancy force and this principle is Archimedes
principle. So, for a floating object, the
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displaced volume is let us call it as V displaced.
So, if on object is floating that means that
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it is not completely sinking into the liquid.
Gravity is acting down, if M is the mass of
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the solid and g is the acceleration due to
gravity, Mg is the weight of the object, solid
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object, weight of the solid. This must be
balanced by V displaced times rho times g.
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Now, the mass of the solid is nothing but,
rho solid times V solid, the volume of the
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solid times g is equal to V displaced time
rho liquid times g. This is condition for
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floating, condition for a body to be at equilibrium
for a body to be floating in a liquid surface.
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So, this is Archimedes principle. What I will
do next is to illustrate this with a straightly
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more involved example. So, let us try to apply
principle for a following problem. You have
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a huge you know swimming pool let us say huge
body of water. This is the water level is
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water and you have an object let say a raft
is floating and over the raft there is a barrel,
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a cylindrical barrel. This is a raft, this
is a barrel. So, this is water so this is
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gravity is acting in this direction and let
us say the displaced volume is V. In this
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case, let us call this situation case A.
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Let us consider two other situations, a second
situation is where you have same barrel and
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raft, but the barrel and the raft are floating
separately. They are partially submerged,
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both are floating. So, let us call the displaced
volume as V raft and this is V barrel. So,
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their floating separately let us call this
case B. And the third a situation is like
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so, you have free surface of the water and
then you have the barrel, but sorry, you have
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the raft and the barrel is such that it is
completely submerged.
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So, let us call this displaced volume. We
are in the both cases of the raft. So, this
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is the displaced volume of the barrel, let
us call this case C. Now, in comparison this
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three cases, what is the level of water of
case B and case C with respect to the lever
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of water in the swimming pool in case A? That
is question that we are going to answer by
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using the Archimedes principle.
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So, let us consider case A. Here, the both
in case A, in both barrel and raft are floating,
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but the displaced volume is only due to the
raft. By using Archimedes principle, the net
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force downwards is given by the sum weights
of the raft and the barrel times g.
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This must be balanced if this combination
is to be floating, then this must be balanced
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by buoyancy upwards. The buoyancy force upwards
which is nothing but, rho water, which I denote
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as rho. This is the density of water
times Vr, let us call it just V that is the
notation we used times g. So, condition for
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the floating in the first case, case A is
Mr plus Mg, Mbg is so rho vg. In case B, barrel
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floats separately, the raft and barrel float
separately. That means that M raft times g
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is V raft times rho g and M barrel times g
is V barrel times rho g. This two must separately
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hold for both this things to float separately.
We can add this two equations Mr plus Mb times
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g is equal to Vr plus Vb times rho g. Let
us call this equation 2, let us call this
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equation 1.
When I compare these two equations, this implies
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that and Vr Mr plus Mb is same on the left
side. So, rho Vg must be equal to the right
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side must also be the same must be equal to
rho Vr plus Vb times g. So, if I cancel rho
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density of water and acceleration due to gravity.
I get that V is Vr plus Vb. That is in case
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B, where the barrel and raft are floating
separately. Here, the net displaced volume
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is the same as the total displaced volume
in case A. So, no change in the water height,
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in the level of water in the swimming pool.
That is the conclusion; we can come to just
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by applying Archimedes principle to these
two cases individually.
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Let us go back to case C. Now, in case C should
remember that the raft is floating. So, if
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the raft is floating then the net weight downwards
must be balanced by buoyancy force acting
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upwards. Rho is a density of water, but this
is raft floating condition, but the barrel
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as sunk. That means that the net downward
weight must be greater than the buoyancy force,
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this is the entire volume of the barrel g.
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So, if I add these two equations then I get
Mr plus Mbg must be greater than or equal
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to rho Vr plus Vbg, let us call this equation
3. Let me rewrite equation 1, which is Mr
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plus Mvg, this is equation 1. That I just
derive few minutes back is equal to rho Vg.
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This means if this is the case, this implies
that if the left side of this is greater than
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this. This means that, this is identically
equal to this. So, rho Vg is greater than;
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so, this is a g here greater than rho Vr plus
Vbg or V is greater than Vr plus Vb. So, this
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is equation 1 from case 1, where this is the
displaced volume when the raft and the barrel
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are floating in this is case A. Now, in the
case C only the raft is floating. This is
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the displaced volume of the raft entire barrel
is sunk.
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So, this implies that the level will fall,
this is case C water level fall in case C
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because displaced volume here is less than
the total volume when the raft and barrel
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both were floating. So, compare to these two
cases this water level will be smaller because
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a displaced volume here is small smaller compared
to the displacement volume in case A. So,
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this is slightly counter intuitive a result
because you have a situation where this barrel
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is completely sunk and you may intuitively
or instinctively think that this water level
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will raise here compare to case here, but
by careful application of Archimedes principle,
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we can show that it is not the case and fact
it is opposite. Now, one last topic fluid
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statics, so far, I have been discussing fluids
under static conditions under the influence
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of gravitational field.
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But in many chemical engineering applications,
you encounter a fictitious body force, call
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the centrifugal force which happens; this
force happens because of rotation. Suppose,
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you have a bowl of, a bowl of liquid like
water and you rotate it. This whole bowl is
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rotated at a high speed. Let us say omega.
Now, if you have liquid in it and it is not
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completely filled, then this liquid the centrifugal
force if you remember from mechanics acts
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from the radius radial access to outwards.
So, this is the direction of centrifugal force.
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So, the centrifugal force on any fluid element
tends to through the fluids towards outside.
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So, you may imagine that if, the bowl is not
fully filled, then this water, initially the
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water level will be like this. It is partially
filled. Now, after rotating this water will
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just be thrown close to the surface and this
situation will happen when gravity is very
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very small, a small compare to the centrifugal
force. When the centrifugal force is so large,
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when the rotation speeds are so large, then
this then this liquid will be completely thrown
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towards the end of the towards the rim of
the container, the bowl. In this case, the
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entire mass of liquid will rotate like a fluid.
Now, imagine this is the axis and this is
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the wall. This is the wall of the bowl and
let us call this radius as r2 and this is
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the interface, this is liquid. So, I am just
trying to blow this region up here and let
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us call the distance of radio distance of
the interface from axis as r1. Now, the centrifugal
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force if you take any tiny fluid element.
Well, it is actually annular, it is a annular
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element. So, because the geometry cylindrical.
So, the fluid element will be annular, it
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is a ring like element. So, if you take any
fluid element there is a unbalance centrifugal
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force acting readily outwards.
So, if the fluid is rotating like a rigid
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object. There is no relative motion between
two fluid elements, then you can think of
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the entire think as so it is a solid like
motion. So, it is under static conditions
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even though it is moving like a rigid body,
there is no relative deformation. So, there
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is no shear stresses. So, the only stresses
are very normal to the fluid elements, so
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the pressure. So, the pressure must be vary
accordingly to balance the body force due
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to centrifugal forces.
So, how does one calculate this? The differential
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force on an element, on a volume element differential
centrifugal force. Centrifugal force on a
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volume element is centrifugal force goes as
mass times the radial distances squared times,
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sorry, the angular velocity square times the
radial distance, Mr omega square. So, if you
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take a differential volume element. So, differential
mass dm times omega square is the angular
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velocity square times r. This is the angle
omega is the angular velocity in radians per
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second. So, this is the mass of the volume,
tiny volume of the differential volume.
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If you take a cylindrical volume element it
is mass is dm is the volume is 2 pi r. So,
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this is r any at any radial distance r. You
consider a slice, cylindrical slice and at
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this height be b or the cylindrical element.
So, 2 pi rb is the surface area of the cylinder
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at in a cylinder r times dr which is a thickness.
This is tiny thickness we are considering.
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This is the volume times the density, this
is the mass this is dm. So, dF is nothing
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but, 2 pi rb times dr rho times omega square
r. This is the differential force on a volume
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00:25:50,869 --> 00:25:57,869
element due to centrifugal forces due to rotation.
Now, this must be balanced by differential
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pressure. So, if you take a tiny slice, a
cylindrical slice there is a differential
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00:26:07,080 --> 00:26:11,929
force that is acting like this. So, the pressure
build up must act such that it balances; this
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00:26:11,929 --> 00:26:17,119
is the pressure acting in this direction.
So, pressure must tend to counteract this
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centrifugal force that acts in this direction.
So, the pressure must balance change in pressure
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must balance this centrifugal force which
tends to; now, let me simplify this slightly.
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So, it is 2 pi r square omega square b rho
dr. That is the force. So, the pressure is
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00:26:47,619 --> 00:26:54,619
force divided by area, area is 2 pi rb. This
is the force pressure is force divided by
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00:26:55,239 --> 00:26:59,109
area. So, let me cancel 2 pi rb.
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So, dp is nothing but, 2 pi b then one r cancels
is rho r omega square dr. Integral dp if I
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want to find the pressure distribution or
difference between any two points and integrate
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this over the two points between p1 and p2,
between any two radial locations r1 and r2.
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00:27:26,119 --> 00:27:33,119
No longer small let us r dr this implies p2
minus p1 is rho omega square by 2 r2 square
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00:27:38,649 --> 00:27:45,649
for minus r1 square. This is a kinetic decision.
So, this is a important result where pressure
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variation is now, a happening in a rotating
fluid where the body forces due to centrifugal
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forces and not due to the gravity. And this
is in the limit when the centrifugal forces
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are large compare to the gravitational forces.
So, we have neglected gravity compare to centrifugal
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forces.
So, just as gravity acts on a objects. So,
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in a gravitational field centrifugal forces
can also act on object and the centrifugal
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00:28:23,749 --> 00:28:30,749
force will tend to accelerate particles with
higher density or element with higher density.
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00:28:30,769 --> 00:28:34,289
And they will be thrown to the wall because
their magnitude of the centrifugal forces
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00:28:34,289 --> 00:28:41,289
are larger and these are used in a many separations.
So, if you want to separate two liquids, two
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immiscible liquid with different densities.
One way is to take this two liquids and put
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00:28:48,289 --> 00:28:53,249
them in a centrifuge which is basically be
a bowl that is rotated with very high speed.
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00:28:53,249 --> 00:28:57,779
And then because of the centrifugal action
the liquid of higher density will be thrown
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00:28:57,779 --> 00:29:03,559
towards the wall and the liquids liquid of
a lower density will be more towards the center.
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00:29:03,559 --> 00:29:08,269
And then you can simply separate this two
away just based on dense difference.
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00:29:08,269 --> 00:29:11,669
The same thing can be done due to gravity
with the help of gravity also. That is called
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00:29:11,669 --> 00:29:18,669
gravity base separation, but the driving force
which is because of acceleration to gravity
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00:29:19,460 --> 00:29:26,460
g is very small. It is fixed rather, but in
case of centrifugal forces we can vary the
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00:29:28,139 --> 00:29:33,820
centrifugal acceleration at our will by changing
the angular velocity of rotation and you can
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00:29:33,820 --> 00:29:40,820
achieve faster rates of separation between
these two fluids. So, this a really completes
225
00:29:40,909 --> 00:29:47,909
my emphasis on fluid statics. So, the next
topic I am going to discuss is Fluid kinematics.
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00:29:59,139 --> 00:30:06,139
So, firstly what is kinematics? In general
in any subjects; any subject that deals with
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00:30:11,840 --> 00:30:17,109
a mechanics whether it is solid mechanics
or fluid mechanics or particle mechanics whatever
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00:30:17,109 --> 00:30:22,779
subject you have it. Mechanics deals with;
as I told in the beginning forces and the
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00:30:22,779 --> 00:30:29,779
motion that cause by forces. So, mechanics
is broadly divided into two parts one is kinematics.
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00:30:30,429 --> 00:30:37,429
Kinematics is the subject that deals with
description of motion without worrying about
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00:30:45,200 --> 00:30:52,200
or without reference to the forces that cause
them. So, that is the first thing is to be
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00:31:04,340 --> 00:31:08,129
able before understanding how forces cause
motion?
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00:31:08,129 --> 00:31:13,269
The first step to first understand how to
describe the motion per say the motion itself
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00:31:13,269 --> 00:31:20,049
and once we have the necessary tools to describe
the motion, then we can go ahead and study
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00:31:20,049 --> 00:31:27,049
how forces cause motion. That subject is called
dynamics. Dynamics relates motion to forces.
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00:31:32,519 --> 00:31:39,519
So, there are these are two branches of mechanics
and so, firstly we will have to understand
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00:31:43,129 --> 00:31:50,129
how to describe fluid motion? So, we will
do kinematics of fluid flows.
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00:31:52,109 --> 00:31:58,720
Because in this course, we are interested
in motion of fluids. So, we will first discuss
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00:31:58,720 --> 00:32:05,720
kinematics of fluid flows, then we will proceed
to dynamics. Now, a useful, so firstly, let
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00:32:08,399 --> 00:32:13,190
us try to understand how to how; what are
the various options we have to describe fluid
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00:32:13,190 --> 00:32:18,200
flows? Now, remember that we have already
taken the continuum route or the continuum
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00:32:18,200 --> 00:32:24,409
approach where in we are saying that the fluid
is a continuous medium. And you can identify
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00:32:24,409 --> 00:32:31,139
each and every point in the fluid and ascribe
unique properties to points such as velocity,
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00:32:31,139 --> 00:32:36,090
pressure, temperature, density and what not.
All kinds of properties can be attributed
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00:32:36,090 --> 00:32:43,090
to each and every point in the fluid and these
various variables such as pressure density
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00:32:44,849 --> 00:32:48,940
and so on. They are smoothly varying functions
of spatial coordinates and time.
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00:32:48,940 --> 00:32:55,940
This is the essential crocks of continuum
hypothesis. Now, how do I, what do I mean
248
00:32:56,679 --> 00:33:03,679
by a point in a fluid? So, in this context
the notion of what is called a hypothetical
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00:33:10,549 --> 00:33:17,549
fluid particle helps in the continuum picture.
What is the fluid particle? Well, imagine
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00:33:20,049 --> 00:33:27,049
let say you have a box containing a liquid
like water and it is stationary initially.
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00:33:29,919 --> 00:33:36,919
Let us say, we take a colored dye and then
mark a fluid here and then take another color
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00:33:37,879 --> 00:33:44,879
dye mark a fluid here. Take another color
dye and mark a fluid here, take a dye of another
253
00:33:45,169 --> 00:33:52,169
color and mark a fluid here and so on.
Imagine that you are marking fluids with colored
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00:33:52,190 --> 00:33:57,960
dye and let us assume that dye molecules do
not diffuse. So, that the dye stays good I
255
00:33:57,960 --> 00:34:02,609
mean it does not if you drop a color. Color
liquid like ink of course, you know that it
256
00:34:02,609 --> 00:34:07,059
is going to dissolve a diffuse and water,
but let us assume the diffusivities are so
257
00:34:07,059 --> 00:34:12,590
small for our timescales of interest here
that you can imagine that the dye does not
258
00:34:12,590 --> 00:34:18,919
dissolve at all. We can dye; imagine the dye
to be insoluble in the liquid that we are
259
00:34:18,919 --> 00:34:25,919
considering. Now, this is the time initially
at time t equal to 0. So, you are imagining
260
00:34:26,720 --> 00:34:31,880
that at each and every point that you can
resolve in your scale of measurements. You
261
00:34:31,880 --> 00:34:36,030
can ascribe you can point a dye and you can
visualize it is motion.
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00:34:36,030 --> 00:34:43,030
So, this is the time t equals to 0. At a later
time all this points may move. For example,
263
00:34:43,370 --> 00:34:49,980
this is blue color dye, let say this blue
color dye element will move. So, let us use
264
00:34:49,980 --> 00:34:56,980
pink color for; at a later time. So, I will
draw the later time picture with a open circle.
265
00:34:58,730 --> 00:35:05,730
So, time t equal to 0 is closed or filled
circles. A later time at time t it is open
266
00:35:16,660 --> 00:35:23,660
circles. That is I am going to indicate the
motion of various points in the fluid and
267
00:35:26,650 --> 00:35:32,520
the fluid is moving due to presumably due
to application of some forces. So, essentially
268
00:35:32,520 --> 00:35:39,520
we have a bunch of points that are marked
by dye, color dye and we are assuming that
269
00:35:40,360 --> 00:35:46,410
the dye molecules do not diffuse and so, that
they just stay put wherever they are and at
270
00:35:46,410 --> 00:35:52,440
time t equals to 0. When there is no force
forcing on the fluid and at a later time due
271
00:35:52,440 --> 00:35:58,000
to application of some forcing the fluid is
under motion and then the fluid is under motion
272
00:35:58,000 --> 00:36:01,760
all these points will move some other points,
some other locations.
273
00:36:01,760 --> 00:36:08,760
All these colored points will move to some
other locations for example, this may go to
274
00:36:08,900 --> 00:36:15,900
here and so on. And red dot, sorry, red dot
will move here. This orange dot will move
275
00:36:19,350 --> 00:36:26,350
here and this yellow dot will move here. So,
these are the locations of these various points.
276
00:36:30,210 --> 00:36:37,210
Notice that I am using close circles for initial
time the location is initial time of various
277
00:36:40,190 --> 00:36:47,190
dye point, dye elements and I am using open
circles for just for the sake of clarity exaggerating
278
00:36:49,640 --> 00:36:56,640
the size of these points, so that you can
see them easily. So, this can happen that
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00:36:57,990 --> 00:37:04,990
various points at were the initially located.
At some locations denoted by close circles
280
00:37:05,360 --> 00:37:11,620
we will move eventually upon fluid motion
to some other locations.
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00:37:11,620 --> 00:37:18,620
Now, these can be thought of as the location
or this let me just. These points can be thought
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00:37:22,960 --> 00:37:29,960
of as fluid particles. Such dyes parts spots
which can be used to identify location of
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00:37:34,370 --> 00:37:41,370
fluids at various points in the fluid at initial
time and the subsequent motion can be thought
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00:37:41,820 --> 00:37:48,820
of as a fluid particle. And within the continuum
hypothesis a fluid element can fluid can be
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00:37:49,950 --> 00:37:56,420
compressed of infinitely a fluid is infinitely
smooth. There is no discreteness in a fluid.
286
00:37:56,420 --> 00:38:01,700
So, you can resolve a fluid to any link scale
you want. So, there are infinitely large numbers
287
00:38:01,700 --> 00:38:07,600
of a fluid particles correspond to each and
every point in space within the continuum
288
00:38:07,600 --> 00:38:12,830
hypothesis.
And just by way of an illustration. I use
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00:38:12,830 --> 00:38:19,830
the notion of coloring the fluid with a dye
element. A dye, a substance to visualize the
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00:38:20,020 --> 00:38:25,480
motion, but idea is you can think of it as
a mathematical framework where at time t equals
291
00:38:25,480 --> 00:38:32,480
to 0. You have various locations in the fluid
which are marked by their initial locations
292
00:38:33,270 --> 00:38:39,350
and at a later time due to application of
forces all these points will start moving
293
00:38:39,350 --> 00:38:43,340
and they will tend to occupy various different
locations in general.
294
00:38:43,340 --> 00:38:50,340
So, this is the notion of a fluid particle.
So, in general at time t you have a fluid
295
00:38:53,680 --> 00:39:00,680
particle which is moving at a later time.
This is time t equals to 0. Initially, that
296
00:39:02,790 --> 00:39:09,790
is there is no motion initially and upon application
of some forcing. This point moves in spatial
297
00:39:12,660 --> 00:39:18,040
coordinate you always have let say an x y
z coordinate with respect to which we are
298
00:39:18,040 --> 00:39:25,040
describe in a motion that that implicit. So,
just to be complete let me just put coordinate
299
00:39:25,100 --> 00:39:31,410
system there. So, there is a point that is
located at time t equals to 0 and it is moving
300
00:39:31,410 --> 00:39:36,800
due to application of forces to some other
location. This is the current location let
301
00:39:36,800 --> 00:39:43,800
us say at time t. This is what is called a
particle trajectory.
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00:39:44,190 --> 00:39:51,190
But, there are infinitely large number of
such points. Let me just show it some other
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00:39:54,250 --> 00:40:01,250
color. So, there are large number of such
points. Because you can assign that to each
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00:40:02,240 --> 00:40:08,490
and every point and space a point and all
these points will start moving upon application
305
00:40:08,490 --> 00:40:15,490
of forces. So, one way to describe fluid motion
is to consider all these points, label all
306
00:40:19,030 --> 00:40:26,030
these points at their initial location before
application of forces. So, let us label all
307
00:40:28,800 --> 00:40:35,800
points in fluid
based on their initial positions.
308
00:40:43,320 --> 00:40:50,320
So, let us call the initial positions of a
point as so, if you put a coordinate system
309
00:40:55,350 --> 00:41:02,350
any point will be denoted by a vector x. So,
this is x y and z coordinate system. So, x
310
00:41:04,090 --> 00:41:11,090
p0, x0 stands for initial time t equals to
0 and p denotes the fluid particles the subscript
311
00:41:17,960 --> 00:41:24,960
p denotes a fluid particle. So, x p0 this
a vector. So, it is comprised of three co-ordinates
312
00:41:32,610 --> 00:41:39,610
x p0, y p0, zp 0 in a Cartesian coordinate
system. So, this is nothing but, the location
313
00:41:48,700 --> 00:41:54,980
of the trajectory of the location of the particles
position of the particle at time t equal to
314
00:41:54,980 --> 00:42:01,980
0 is x p0. This is the initial location of
various particles, various fluid particles.
315
00:42:14,440 --> 00:42:21,440
Now, upon influence of forces this the location
of these particles at time t will change in
316
00:42:30,180 --> 00:42:37,180
general of various particles will change due
to fluid motion. Now, this x p of t the current
317
00:42:45,230 --> 00:42:52,230
location is a function of the initial locations.
So, once I tell you what are the initial locations
318
00:42:54,680 --> 00:43:00,080
various particles, the current location of
the various particles will be a function of
319
00:43:00,080 --> 00:43:05,980
the initial location. Because a point that
was here will move here, a point that was
320
00:43:05,980 --> 00:43:11,710
here, will move to some other location. This
point can never move here, unlike this point
321
00:43:11,710 --> 00:43:18,230
will never move here in a given set of fluid
p in a specified flow. Each point which started
322
00:43:18,230 --> 00:43:25,230
out initially, at x p0 will eventually, lead
reach a unique x p at time t. So, the current
323
00:43:26,830 --> 00:43:33,830
position of various fluid particles fluid
particles will be function of the initial
324
00:43:40,410 --> 00:43:47,410
positions.
Now, if I know this functional form sometimes
325
00:43:51,830 --> 00:43:58,830
I just people just write this as x p of t
is x p of x p naught and time that is the
326
00:44:01,520 --> 00:44:08,520
function itself is denoted by the same symbol.
Now, if I have this functional form how the
327
00:44:09,660 --> 00:44:15,000
current portion varies with initial portion
and time then I can calculate the velocity
328
00:44:15,000 --> 00:44:22,000
of a fluid particle at time t. Velocity of
a fluid particle x p naught at time t a fluid
329
00:44:27,300 --> 00:44:32,650
particle is identified by its location at
time t equals to 0. So, we use some other
330
00:44:32,650 --> 00:44:38,690
color here. At time t equal to 0, a fluid
particle is at x p naught and this particle
331
00:44:38,690 --> 00:44:45,690
moves here, some other particle which was
here would move there at time t. So, each
332
00:44:47,050 --> 00:44:54,050
particle, this particle time t here is labeled
by it is location at time equals to 0. So,
333
00:44:59,230 --> 00:45:02,740
each particle is labeled by their initial
locations.
334
00:45:02,740 --> 00:45:09,630
So, this is the velocity of the particle which
was at time x p0 at time t equal to 0 and
335
00:45:09,630 --> 00:45:13,360
the velocity of that particle at time t is
nothing but, the rate of change of its position
336
00:45:13,360 --> 00:45:20,360
dx p by dt keeping x p0 constant. That is
your fixing the same particle, that you are
337
00:45:22,760 --> 00:45:26,950
following the same label in some sense and
you are asking what is the velocity at time
338
00:45:26,950 --> 00:45:32,950
t valid is simply the rate of change of it
is position vector which is x p. Now, such
339
00:45:32,950 --> 00:45:39,950
a description is called; such description
of fluid flow where in your identifying various
340
00:45:41,610 --> 00:45:47,370
particles by their locations at time t equals
to 0. And merely following the positions of
341
00:45:47,370 --> 00:45:52,020
various particles as a function of time is
called the Lagrangian description.
342
00:45:52,020 --> 00:45:59,020
It is called the Lagrangian description in
fluid flow. What is the Lagrangian description?
343
00:46:02,280 --> 00:46:09,280
Well, here the independent variables are the
position the initial position of various particles
344
00:46:15,710 --> 00:46:22,710
that is the label and time. So, all properties
such as suppose velocity of a particle is
345
00:46:23,760 --> 00:46:28,990
a function of it is initial position and time.
So, what is this? This is the velocity of
346
00:46:28,990 --> 00:46:35,920
a particle which was at x p0 at time t equal
to 0. At and as it particle moves at a later
347
00:46:35,920 --> 00:46:42,920
time what is it is velocity? You are following
various particles and then your enquiry what
348
00:46:43,170 --> 00:46:47,480
is the velocity? What is the acceleration?
What is the density? What is a pressure? What
349
00:46:47,480 --> 00:46:51,140
is their temperature and so on?
350
00:46:51,140 --> 00:46:57,610
So, independent variables in a Lagrangian
description are, so, if you have any function
351
00:46:57,610 --> 00:47:04,060
any property it is given as a function of
the initial position of the particle which
352
00:47:04,060 --> 00:47:11,010
is a essentially serving as the label of the
particle and time. Now, so not just for flow
353
00:47:11,010 --> 00:47:18,010
variables, even if can think of temperature
in a fluid, in a moving fluid. This is a temperature
354
00:47:19,040 --> 00:47:26,040
of a particle which was at a time t equal
to 0 at x p0 at a later time t. So, various
355
00:47:26,720 --> 00:47:32,510
particles as they move their property such
as density, temperature, concentration and
356
00:47:32,510 --> 00:47:39,160
velocity and many other things will change,
but you can describe the change based on their
357
00:47:39,160 --> 00:47:42,720
initial co-ordinates. This is the essential
idea behind Lagrangian description.
358
00:47:42,720 --> 00:47:49,720
So, once I have this motion, how the trajectory
of the particle changes various particles
359
00:47:55,820 --> 00:48:02,820
change as function of time? Then I get velocity
of a particle which was at x p0 at time t
360
00:48:03,670 --> 00:48:10,670
equal to 0. At a later time t is nothing but,
the rate of change of it is position by keeping
361
00:48:12,910 --> 00:48:19,910
x p0 same that is your following the same
particle. Now, acceleration of a particle
362
00:48:24,030 --> 00:48:31,030
is nothing but, the rate of change of its
velocity keeping the same particle such a
363
00:48:34,460 --> 00:48:41,460
description is also called sometime as the
Material description or the Lagrangian description.
364
00:48:50,200 --> 00:48:57,200
Now, this is a one of way of describing the
fluid motion, but this is not the only way
365
00:49:00,270 --> 00:49:04,710
or this is neither is this is the most useful
way. What is normally done in fluid mechanics
366
00:49:04,710 --> 00:49:11,710
is what is called the Eulerian description?
So, what is a Eulerian description? Here,
367
00:49:20,840 --> 00:49:27,840
we place a lab coordinate frame in our lab
x, y, z and measure various properties such
368
00:49:29,350 --> 00:49:36,350
as velocity in the fluid as a function of
three spatial laboratory coordinates and time.
369
00:49:37,800 --> 00:49:44,560
So, velocity is measured not as a function
of various particles not by following the
370
00:49:44,560 --> 00:49:51,560
particles fluid particles.
But by merely saying what is suppose I put
371
00:49:51,710 --> 00:49:58,520
flow measuring velocity meters at speed various
points in space and then you measure velocity
372
00:49:58,520 --> 00:50:03,690
at each and every point as a function of time.
So, that description is the Eulerian description.
373
00:50:03,690 --> 00:50:10,150
This is also called a Spatial description
for obvious reasons because we are describing
374
00:50:10,150 --> 00:50:17,150
a part, a properties of the fluid such as
velocity and acceleration, pressure, density,
375
00:50:19,170 --> 00:50:23,960
temperature as a function of by following
the various particles. So, in the Lagrangian
376
00:50:23,960 --> 00:50:29,400
description by labeling a particle, when I
say temperature of a particle, what I mean
377
00:50:29,400 --> 00:50:33,260
is that as I follow the particle how is the
temperature changing with time. That is the
378
00:50:33,260 --> 00:50:37,030
Lagrangian description.
In the Eulerian description, we are not following
379
00:50:37,030 --> 00:50:41,910
the particle any more, fluid particles any
more. We are simply keeping a stationary frame
380
00:50:41,910 --> 00:50:46,420
of reference, a lab frame of reference and
this could be stationary or it could move
381
00:50:46,420 --> 00:50:51,680
with a constant velocity that depends on the
nature of the problem. For simplicity, let
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us keep a fixed coordinate system in our lab.
And then we can measure various quantities
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such as velocity or temperature or pressure
at various points with respect to this coordinate
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system as a function of time. And report quantities
like how does a temperature change at various
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points in the fluid as a function of time.
So, what is T of x, t? At a given location,
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if I fix x. So, if I have a 3 Cartesian co-ordinate
system. I can fix x. At a fixed x, how does
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a temperature change as a function of time?
Or at a given time how does a temperature
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vary as a function of x, y and z? So, this
is called the spatial description in fluid
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mechanics. Now, such quantities are called
fields. This is called the velocity field,
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where the velocity is expressed as a function
of a 3 spatial coordinates and time. This
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is called the temperature field.
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Now, what this description does is that suppose,
I put a thermometer; suppose, I have fluid
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that is flowing, I put the thermometer. So,
this a thermometer and I measure T at a given
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location let us call this x and time. What
this is measuring is at a time T is measuring;
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whatever so, the fluid is continuously flowing.
A fluid particle with will occupy this spatial
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location x at a time T and T of x, t will
be the temperature of that fluid particle
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denoted by this pinkish violet circle which
happens to occupy this spatial location at
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time t.
At a later time, t plus delta t a slightly
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later time at the same spatial location. This
point would have moved let us say here, this
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point which was here at x, that are moved
here and some other point would come and occupy
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the location x. This is the same location
x, so, let us call this vector x. So, different
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fluid particles the key to understand here
is the different fluid particles will occupy
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spatial positions at the same spatial position
at different times by virtue of fluid motion.
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Because a fluid is continuously moving or
flowing. So, what the thermometer will measure
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at a given spatial location it is merely a
record of the temperature values at that location
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as a function of time. And this does not correspond
to the temperature of fluid particles. So,
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the Eulerian description gives a very very
completely different view point compared to
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Lagrangian description. In the Lagrangian
description you would follow the same particle
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as a function of time.
So, whereas in the Eulerian description, you
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are not following the same particle. You are
merely sitting at a same point and space and
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you are recording various properties such
as temperature in this particular instance.
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So, we are measuring different the properties
of various fluid particles not the same fluid
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particle. So, we will stop here and will continue
in the next lecture further. We will see you
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in the next lecture.