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Everything we've done so far has
been stationary fluids, or

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static fluids, and we've been
dealing with static pressure.

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We were trying to figure out
what happens when everything's

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in a steady state.

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Now let's work on what
happens when the

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fluid is actually moving.

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Let's imagine a pipe.

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Let's say one end of the pipe
has a larger area than the

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other end, or at least
a different area.

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So this is one end of the pipe,
and this is the other

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end of a pipe.

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It's filled with some fluid,
some liquid, actually, in our

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example, so there's just a bunch
of liquid in this fluid.

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Let's say this area
at the entrance is

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called the area in.

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That's the area of the opening
into the pipe, and let's call

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this area out.

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It's the area of the opening
coming out of the pipe.

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Let's think about what
happens if this

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liquid is actually moving.

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Let's say it's moving into the
pipe with the velocity V in.

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Let's think about how much
volume moves into the pipe

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after T seconds.

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After T seconds, if you
think about it, you'd

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have this much area.

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If you think about what was
right here, it will then be

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moved to the right
by how much?

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We could just go back to our
basic kinematic formula:

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distance is equal to
rate times time.

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The distance something travels
equals velocity times time, so

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after T seconds, whatever fluid
was here, it would have

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an area of about that much.

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Whatever fluid was there
would have traveled

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how much to the right?

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It would have traveled-- let's
assume that the pipe doesn't

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change too much in diameter or
in radius from here to here.

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It would have traveled velocity
times time, so V in

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times time.

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It could be meters or whatever
our length units are.

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After T seconds, essentially
this much water has traveled

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into the pipe.

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You could imagine a cylinder
of water here.

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Once again, I know I made it
look like it's getting wider

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the whole time, but let's assume
that its width doesn't

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change that much over the T
seconds or whatever units of

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time we're looking at.

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What is the volume of this
cylinder of water?

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The volume-in over the T seconds
is equal to the area,

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or the left-hand side
of the cylinder.

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Let me draw the cylinder in a
more vibrant color so you can

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figure out the volume.

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So it equals this side, the left
side of the cylinder, the

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input area times the length
of the cylinder.

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That's the velocity of the
fluid times the time that

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we're measuring, times the input
velocity times time.

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That's the amount of volume
that came in.

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If that volume came into the
pipe-- once again, we learned

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several videos ago that the
definition of a liquid is a

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fluid that's incompressible.

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It's not like no fluid could
come out of the pipe and all

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of the fluid just
gets squeezed.

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The same volume of fluid would
have to come out of the pipe,

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so that must equal
the volume out.

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Whatever comes into the pipe has
to equal the volume coming

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out of the pipe.

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One assumption we're assuming in
this fraction of time that

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we're dealing with is also that
there's no friction in

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this liquid or in this fluid,
that it actually is not

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turbulent and it's
not viscous.

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A viscous fluid is really just
something that has a lot of

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friction with itself and that
it won't just naturally move

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without any resistance.

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When something is not viscous
and has no resistance with

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itself and moves really without
any turbulence, that's

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called laminar flow.

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That's just a good word to
know about and it's the

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opposite of viscous flow.

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Different things have different
viscosities, and

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we'll probably do
more on that.

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Like syrup or peanut
butter has a

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very, very high viscosity.

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Even glass actually
is a fluid with a

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very, very high viscosity.

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I think there's some kinds of
compounds and magnetic fields

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that you could create that have
perfect laminar flow, but

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this is kind of a perfect
situation.

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In these circumstances, the
volume in, because the fluid

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can't be compressed, it's
incompressible, has to equal

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the volume out.

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What's the volume out over
that period of time?

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Similarly, we could draw this
bigger cylinder-- that's the

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area out-- and after
T seconds, how much

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water has come out?

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Whatever water was here at the
beginning of our time period

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will have come out and we can
imagine the cylinder here.

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What is the width
of the cylinder?

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What's going to be the velocity
that the liquid is

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coming out on the
right-hand side?

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Capital V is volume, and
lowercase v is for velocity,

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so it's going to be the output
velocity-- that's a lowercase

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v-- times the same time.

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So what is the volume that has
come out in our time T?

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It's just going to be this area
times this width, so the

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output volume over that same
period of time is equal to the

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output area of this pipe
times the output

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velocity times time.

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Once again, I know I keep saying
this, but this is kind

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of the big ah-hah moment, is
in that amount of time, the

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volume in this cylinder
has to equal the

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volume in this cylinder.

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Maybe it's not as wide, or
something like that, but their

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volumes are the same.

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You can't get more water here
all of a sudden than what's

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going in, and likewise, you
can't put more water into the

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left side than what's coming out
of the right side, because

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it's incompressible

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These two volumes equal each
other, so we know the area of

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the opening onto to the left
hand of the pipe times the

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input velocity times the
duration of time we're talking

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about is equal to the output
area times the output velocity

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times the duration of time
we're talking about.

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It's the same time on both sides
of this equation, so we

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could say that the input area
times the input velocity is

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equal to the output area times
the output velocity.

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This is actually called in fluid
motion the equation of

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continuity, and it leads to
some interesting things.

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We'll do some problems
with it in a second.

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One thing that I want to
introduce at this point as

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well is what is the
volume per second?

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Because this is also something
we're going to deal with in a

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second, probably in the next
video, because I'm about to

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run out of time.

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We said that in T seconds we
have this amount of volume

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coming in and it's the same
coming in as coming out.

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So what is the volume
per second?

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It's this big capital Vi
per amount of time, and

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we call that flux.

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We'll learn a lot about flux,
especially when we start doing

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vector calculus, but flux is
just how much of something

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crosses a surface in
an amount of time.

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It's how much a volume
crosses a surface in

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an amount of time.

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So in this case, the
surface is the

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left-hand side of cylinder.

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And we're saying how much
crosses in amount of time?

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We figured out it's that input
volume, which crosses in every

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T seconds, and this
is called flux.

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You've probably heard of the
flux capacitor in Back To The

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Future, and maybe we can think
about what they were

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trying to hint at.

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Let's see if we can use flux
and these ideas to come up

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with some other interesting
equations.

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We know that the volume per
time is equal to flux.

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This is a big V.

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V is equal to flux, and actually
the variable people

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generally use for flux is R.

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Of course, it's in meters
cubed per second.

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That's its unit.

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We also know that the input area
times input velocity--

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that's a lowercase v-- is equal
to the output area times

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output velocity, and this is
called the equation of

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continuity.

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It holds true whenever
we have laminar flow.

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Actually, I'm about to
run out of time.

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In the next video, I'm actually
going to use some of

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this information to figure out
how much power is there in a

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system where we have fluid
going through a pipe.

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See you soon.

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