Bernoulli's equation of total energy

0:01Let'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?

0:25You 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?

0:42And 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?

0:58That 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.

1:14So 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.

1:35So 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.

2:06OK, 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.

2:27But 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.

2:57And 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.

3:27And 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.

3:46So 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.

4:05And 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.

4:26The 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.

4:59And 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.

5:32And 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.

6:03And 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.

6:25And 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.

6:45And 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.