1
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2
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Let's say I have a horizontal
pipe that at the left end of

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the pipe, the cross-sectional
area, area 1, which is equal

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to 2 meters squared.

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Let's say it tapers off so that
the cross-sectional area

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at this end of the pipe,
area 2, is equal to

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half a square meter.

8
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We have some velocity at this
point in the pipe, which is

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v1, and the velocity exiting
the pipe is v2.

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The external pressure at this
point is essentially being

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applied rightwards
into the pipe.

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Let's say that pressure
1 is 10,000 pascals.

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The pressure at this end, the
pressure 2-- that's the

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external pressure at that point
in the pipe-- that is

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equal to 6,000 pascals.

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Given this information,
let's say we have

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water in this pipe.

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We're assuming that it's laminar
flow, so there's no

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friction within the pipe, and
there's no turbulence.

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Using that, what I want to do
is, I want to figure out what

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is the flow or the flux of the
water in this pipe-- how much

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volume goes either into the pipe
per second, or out of the

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pipe per second?

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We know that those are the going
to be the same numbers,

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because of the equation
of continuity.

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We know that the flow, which
is R, which is volume per

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amount of time, is the same
thing as the input velocity

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times the input area.

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The input area is 2, so it's
2v1, and that also equals the

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output area times output
velocity, so it equals 1/2 v2.

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We could rewrite this, that v1
is equal to 1/2 R, and that v2

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is equal to 2R.

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This immediately tells us that
v2 is coming out at a faster

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rate, and this is based on
the size of the openings.

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We know, because V2 is coming
out at a faster rate, but we

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also know because we have much
higher pressure at this end

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than at this end, that the water
is flowing to the right.

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The pressure differential, the
pressure gradient, is going to

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the right, so the water
is going to

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spurt out of this end.

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And it's coming in this end.

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Let's use Bernoulli's equation
to figure out what the flow

46
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through this pipe is.

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Let's just write it down: P1
plus rho gh1 plus 1/2 rho v1

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squared is equal to
P2 plus rho gh2

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plus 1/2 rho v2 squared.

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This pipe is level, and the
height at either end is the

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same, so h1 is going
to be equal to h2.

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These two terms are going to be
equal, so we can cross them

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out-- we can subtract that value
from both sides, and

54
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we're just left with P1.

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What's P1?

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P1 is 10,000 pascals plus 1/2
rho times v1 squared.

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What's v1?

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That's R over 2-- we figured
that out up here.

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v2 times R over 2 squared is
equal to P2, and that's 6,000

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pascals plus 1/2 rho
times v2 squared.

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We figured out what v2 is--
v2 is 2R squared.

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Let's just do some
simplification, and so let's

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subtract 6,000 from both sides,
and we're left with

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4,000 plus rho R squared over
8 is equal to 1/2 times R

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squared times 4.

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So this is 2 rho R squared.

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We could multiply both sides of
this equation by 8, just to

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get rid of this in the
denominator, so we would get

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32,000 plus rho R squared is
equal to 16 rho R squared.

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Subtract rho R squared from both
sides of this question,

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and we get 32,000 is equal
to 15 rho R squared.

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Then what's rho?

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What's the density of water?

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The density of water is 1,000
kilograms per meter cubed, so

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this is 1,000.

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Let's divide both sides
by 15 times rho.

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We get R squared is equal to
32,000 divided by 15 rho-- rho

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is 1,000, so R squared is equal
to 32,000 over 15,000,

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which is the same thing
is 32 over 15.

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R is equal to the square root
of 32 over 15, and that's

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going to be meters
cubed per second.

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I get 32 divided by 15 is equal
to 2.1, and the square

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root of that 1.46.

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So the answer is R is equal to
1.46 meters cubed per second.

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That is the volume of water that
is either entering the

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system in any given second, or
exiting the system in any

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given second.

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We can figure out the
velocities, too-- what's the

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velocity exiting the system?

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What's two times that?

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It's 2.8 meters per second
exiting the system, and going

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in it is half that, so it's
0.8 meters per second.

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Hopefully, that gives you--
actually, 0.7 meters per

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second-- a bit more intuition on
fluids, and that's all I'm

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going to do for today.

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I'll see you in the next video,
and we're going to do

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some stuff on thermodynamics.

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