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## Fluid Dynamics

# Bernoulli's equation derivation part 2

## Video transcript

This is just a quick review
of what we were doing in the last video. We had this oddly shaped pipe
and the fluid coming in had input velocity V1. The pressure on the left-hand
side pushing to the right is P1, and the area of
this hole is A1. Everything that the same
variables with the 2 on it is coming out of the pipe. What we just set up in that last
video is we said by the law of conservation of energy,
essentially the joules, the energy at this point in the
system, or that we're putting into the system, has to be equal
to the energy coming out of the system. We used that information to set
up this big equation, but it's not too complicated. We figured out that the work
going into the system was the input pressure times the mass of
volume over some period of time divided by the density of
whatever type of liquid we had, which was the
potential energy. This is just typically mgh,
where the mass is the mass of this column of fluid. We're saying, how much work
was done over some period of time T? That's the way I would
think about it. How much energy was there over
some period of time T? The kinetic energy over that
period of time would have been the mass of this volume of
fluid times its velocity squared divided by 2. That's typical kinetic energy. Of course, that has to be
equal to essentially the output energy, and so this is
the output work, or how much work a column of water could
do on the output side. It's an equivalent volume
of water, remember that. In some period of time T,
whatever volume of water this was, an equivalent volume of
water-- maybe it'll be a longer cylinder now, because
it's going to be going faster. So on the output side, it's
this longer cylinder that we're talking about, but it's
going to be the same volume and the same mass. So what we say is that the work
that this column can do in that same amount of time
would be the output pressure times the mass of this column
divided by the density of the column-- which is the same
because the density of the liquid is the same throughout--
times the mass of this column, which is the same
as the mass of this column because the volume and the
density hasn't changed, so they're the same mass. Now, this column has more
potential energy. It's up at h2, which I'm
assuming is higher than h1. This kinetic energy is just the
mass of this cylinder of fluid times its velocity
squared, which is the output velocity divided by 2. This is potential energy
out, and this is kinetic energy out. These equal each other. This setup is Bernoulli's
equation, but let's see if we can clean it up so that we can
get rid of variables that we don't have to know about. One thing that we see is that
there's an m in every term, so let's get rid of them. Divide both sides of
this equation by m. We get that. I don't like this density in the
denominator here, so let's multiply both sides of this
equation by density, and what we're left with is-- let me
write this in a vibrant color. P, the input pressure, plus--
and we're multiplying everything by this rho,
this density. So we have input pressure plus
rho g h1, the input height, the initial height, plus
rho v squared over 2. This is rho v squared over
2, and that equals-- we multiplied both sides by rho, so
we get the input velocity, so that equals the pressure
out plus the density times gravity times the
output height. Let's make everything
consistent. I wrote 2's here, so let's just
say this is pressure 2, this is height 2, plus rho times
the velocity squared. This is Bernoulli's equation,
and it has all sorts of what I would say is fairly neat
repercussions. For example, let's assume that
the height stays constant, so we can ignore these middle
terms. If the height is constant, if I have a higher
velocity and this whole term is constant, then my pressure
is going to be lower. Think about it: If height is
constant, this doesn't change, but if this velocity increases,
but this whole thing is constant, pressure
has to decrease. Similarly, if pressure
increases, then velocity is going to decrease. That might be a little
unintuitive, but the other way, it makes a lot of sense. When velocity increases, this
pressure is going to decrease, and that's actually what makes
planes fly and all sorts of neat things happen,
but we'll get more into that in a second. Let's see if we can use
Bernoulli's equation to do something useful. You should memorize this,
and it shouldn't be too hard to memorize. It's pressure, and then you
have this potential energy term, but instead of mass,
you have density. You have this kinetic
energy term. It's not kinetic energy
anymore, because we manipulated it some,
but instead of mass, you have density. With that said, let's
do a problem. I'll keep this down here, since
you probably haven't memorized it as yet. Let me erase everything else. That's not how I wanted
to erase it. That's how I wanted
to erase it. I wanted to erase it like that
without getting rid of anything useful. OK, that's good enough. And then let me clean up. Clean up all this stuff. Let's say that I have a cup. I'll just draw a cup. It's easier to draw sometimes
then to draw straight lines and all of that. No, that's too dark. Do purple. I'm using a super-wide tool. I have to switch the length. OK, so that's my cup. It has some fluid. Actually, let's say it has a
top to it, and I have some fluid in it. Maybe it happens to be red. We haven't been dealing with
red fluids as yet,. Let me-- oh, I didn't
want to do that. So you know there's
a fluid there. And let's say that there's no
air here, so this is a vacuum. Let's say that h-- we don't know
what units are, but let's say h meters below the
surface of the fluid. This is all fluid here. I poke a hole right there, and
fluid starts spurting out. My question to you is, what is
this output velocity of the fluid as a function
of this height? Let me tell you something
else. Let's say that this hole is so
small, let's call the area of that hole A2, and let's say that
the surface area of the water is A1. Let's say that hole is so small
that the surface area the water-- let's say that A2
if equal to 1/1,000 of A1. This is a small hole relative
to the surface area of this cup. With that said, let's see what
we can do about figuring out the velocity coming out. Bernoulli's equation tells us
that the input pressure plus the input potential energy plus
the input kinetic energy is equal to the output,
et cetera. So what is the input pressure? Well, the input pressure, the
pressure at this point, there's no air or no fluid above
it, so the pressure at that point is zero. What is the input height? Let's just assume that the hole
is done at height 0, h equals 0, so the input
height h1 is just h. If this is 0, then this height
right here is h. What is the input velocity? We know from the continuity
equation, or whatever that thing was called, that the input
velocity times the input area is equal to the
output velocity times the output area. We also know that the output
area is equal to 1/1,000 of the input area-- and this is
area 2-- so we know that the input velocity times area 1 is
equal to the output velocity times 1/1,000 of area 1. We could say area 1 over
1,000 and divide both sides by area 1. We know that the input velocity
is equal to V2 over 1,000, so that's good to know. These are the three inputs into
the left-hand side of Bernoulli's equation. What's on the right-hand side
of Bernoulli's equation? What's P2? What's the pressure
at this point? Oh, I just ran out of time. I'll continue this into
the next video.