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# Bernoulli's equation of total energy

Video transcript

Let's say you're looking
at this blood vessel, and the first thing
you obviously notice is that it's full of
atheromatous plaque. That's what all this
yellow and white stuff is. This is atheromatous plaque. And your thought,
other than, wow, that's a lot of
fatty meals someone's been having, is
how does blood move through this narrow
little space? You got a little
channel that the blood is supposed to be
moving through. How does it get through there? So you do a little experiment. You take a few
pressure readings. You say, OK, let's
figure out what the pressure is right here. And it's about 90. And you say, what
is it right here? And it's 70 in the
middle of the channel. And you say, what is
it on the other side? And over here it's about 80. So you're looking here, and
you're saying, OK, 90 to 70, that makes sense. That blood is moving from
high to low pressure. But then what's going
on between 70 and 80? That seems kind
of strange, right? Because we usually think
of pressure as going from-- or blood moving from
high to low pressure. And here blood is moving
from 70 and then going to 80. And this seems a
little counterintuitive because this is going against
the pressure gradient. So how is that possible? Or have we made a mistake? So to answer that question, we
turn to a Swiss mathematician. This guy came up with
a set of formulas that helps us frame how
we think about this issue, and his name is Bernoulli. So you might have heard
of Bernoulli's equation. So Bernoulli's equation
basically looks like this. He says total fluid energy
equals a few things. It's not just pressure, but
it's pressure plus, let's say, kinetic energy. And I'll explain what all of the
symbols mean in just a second. There we go. So he said the big P
is pressure energy. OK, well, that
part we understood. We were already
looking at pressure and thinking about
why it is that it's going from a low pressure
to high pressure. But he said you also have
to look at movement energy. This is movement energy,
and another word for that would be kinetic energy. But the little p
right here is density, the density of
whatever fluid it is. Here we'd be
talking about blood. And v is the interesting one. This is the velocity, how
fast the blood is moving. So now we have to
actually consider how quick the blood is moving. And then he also talked
about a third term-- this is here-- which
is potential energy. And here he's talking
about the potential energy as it pertains to gravity. So g is gravity. The little p again is density. Then we've got gravity. And we've got height, how high
something is off the ground. So here he's saying
if you have some blood in your head,
obviously that's going to be higher off the ground
than blood in your toe. And there's some
potential energy that comes with being in your
head versus being in your toe. And so that's what
that potential energy part is talking about. Now, for our example, I'm going
to go ahead and erase that. And you'll see why,
because really, the height of all three, I'm assuming,
is at the same level. So there should be
really no difference between the potential energy
from a height standpoint for the points that I
have shown in my picture. So really I'm left
with just that. So if I'm going to
try and figure out the answer to my
problem, I think it would be helpful
to use this equation. And let's see how we can use it. So to figure this
out, let's call this A and let's call
this B, this point. Now, what Bernoulli
wanted to say is that pressure and movement
energy, in this case, combine to stay
the same over time. So A and B have the
same total energy. The total energy remains the
same between the two points. The total energy at A
equals total energy at B. And if we think about
it that way, then you actually can easily
figure out what is going on. I'll show you what I mean. So total energy of A is going
to be the 70 pressure plus 1/2 the density of blood times
the velocity at A squared. And that's got to equal 80
plus 1/2 density of blood, the velocity of B squared. So if this number
right here is smaller-- and it is-- than
the 80, and that's where the whole
problem started with, and we know that overall this
has got to be the same as this, well, then the only
explanation would be that this term right
here has got to be bigger. And there's no other
way to explain this. And Bernoulli was right, that
if you actually look and check the velocity of
blood, how fast it's moving, when it goes through
little tiny channels, like let's say you have
a skinny, little gap between this point
over here, right here, and this point
right here, when it's trying to get
through a little gap, it doesn't have much
space to move through. And so when blood is
moving through tiny spaces, it has to speed up. And that makes complete
sense because we have a large amount
of blood we need to move from here all
the way over here. And the only way to get
all that blood through is that when we have
less space to do it with, to move it even quicker,
to make it go even faster. And so as it's going through
this skinny little channel, the velocity goes way up. So that's where this starts
to really fit together. So this is going way up. And that makes sense because
the density stays the same. So the only difference
is that the velocity of A goes way, way up. And that explains why
you have less pressure at point A versus point B,
but overall total energy at point A and B are the same.