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Bernoulli's equation of total energy

Learn how total energy of a fluid helps explain why fluids can move from low pressure to high pressure! Rishi is a pediatric infectious disease physician and works at Khan Academy. Created by Rishi Desai.

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  • blobby green style avatar for user zunxu tian
    I remembered you did a video said that the smaller the diameter of the artery or tube is the higher the pressure . Why in this video is the opposite ?
    (14 votes)
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    • piceratops ultimate style avatar for user fred
      The smaller the diameter of the artery or tube is the higher the pressure CHANGE, or pressure DROP, that is, the pressure difference measured from before entering the stricture and after will be greater if a tube has small diameter.
      (5 votes)
  • blobby green style avatar for user David Behrend
    I have a similar question as others have in this forum, which I don't think has been adequately answered; in this video pressure is lower in the part of the tube (artery) with a lower diameter. This directly seems to contradict earlier videos about resistance in tubes (in the blood pressure series). In those videos, as diameter of the tube decreases, resistance increases, which causes pressure (specifically change in pressure) to increase, via the equation: deltaP = Q * R. In this equation, R = resistance, which is inversely related to the diameter of the tube. For example, vasoconstriction decreases an artery's diameter, which increases blood pressure. Please help clarify this for me and others.
    (6 votes)
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    • blobby green style avatar for user kaltwater
      I did not see this video but can answer your question. If there was only a single continuous tube then yes as the radius decreases the pressure would increase (due to increased resistance). However, the key is that the big vessels split into many smaller tubes so there are parallel pathways that the blood can flow. When you add parallel pathways you decrease the overall resistance of the system. 1/R(total) = 1/R1 + 1/R2 + 1/R3 .... and so on. You can look up the formula if you have any questions. This is why in capillaries which are very small diameter vessels - the pressure is very low. Also this is why when you remove an organ you increase the resistance of the whole system which is sometimes a question on tests.

      I'm an MD- learned it in med school.
      (4 votes)
  • marcimus pink style avatar for user Sammy Rea (>")> <("<) <3
    I don't get why the pressure is lower in and area where there is littler amount of space to get thought
    (5 votes)
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    • hopper jumping style avatar for user Ravi
      I think you are confusing with concept related to static fluid where pressure is related inversely to volume that it occupies. This is a moving fluid where bernouli sheds light on relation between pressure and area or velocity
      (3 votes)
  • piceratops ultimate style avatar for user Girl on Fire
    Why does 'little p' stand for density? Shouldn't it be a little D?
    (3 votes)
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  • blobby green style avatar for user friaseric90
    I know this question may sound silly, but I am having a hard time reconciling the Venturi Effect as described by Bernoulli's principle and Vasoconstriction/Vasodilation. So according to Bernoulli, in constant fluid flow, as fluid moves from a broader tube to a narrower tube, there is a pressure decrease and subsequent increase in fluid velocity in order to conserve the principle of continuity. So basically, we see a drop in pressure with a narrower tube. However, in blood fluid dynamics, we see an increase in blood pressure in vasoconstriction where the blood vessels become narrower, which appears to contradict Bernoulli's principle in the Venturi effect. I know I am probably applying something incorrectly but I was wondering if anyone could clarify this for me.
    (5 votes)
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  • piceratops seedling style avatar for user seosol21
    Wow.. I absolutely can't understand this video.. hahahaha
    As someone already asked below, smaller diameter of artery has higher pressure in the previous video. Someone answered that high pressure exists before entering the tube and after.. right? So.. this means that pressure in the middle of artery is low? And why pressure increases after if blood came out from small diameter to wider side? I really cannot undersatnd what is going on here. Somebody help me!!
    (3 votes)
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  • piceratops ultimate style avatar for user Ari Mendelson
    What kind of experiments did Bernoulli do to discover this equation?
    (4 votes)
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  • female robot grace style avatar for user SarahLulu
    How does the fluid energy just before the constricted area compare to just after? If they are the same, Bernoulli's equation suggests that because Pressure decreases from 90 to 80 units, the fluid velocity must increase slightly. But why would the velocity change if they are both of the same diameter? If it does change, why would it increase?
    (3 votes)
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  • blobby green style avatar for user abuslik
    In this video the pressure at the inlet is 90 units, in the center it is 70 units, and at the exit it is 80 units. But the diameter at the inlet and exit are the same, so according to Bernoulli's equation the pressure at the exit should be 90 units. What is missing is the frictional losses. How can this be added to Bernoulli's equation most simply?
    (3 votes)
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  • blobby green style avatar for user Somi
    Hi, does Bernoulli's equation only apply to arterial circulation or does it work for venous return?
    (2 votes)
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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.