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Resistance in a tube

Discover the physics behind fluid resistance! Learn how Dr. Jean Louis Marie Poiseuille's 19th-century equations shed light on why it's harder to blow air through a straw than a tube. Explore how this principle applies to blood flow in our bodies, impacting resistance in our blood vessels. Rishi is a pediatric infectious disease physician and works at Khan Academy. Created by Rishi Desai.

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  • leafers sapling style avatar for user Peter Collingridge
    What's the reason for r being raised to the fourth power? I can see why it would be squared (to get the cross sectional area of the tube), but I can't see why it's squared again.
    (30 votes)
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    • duskpin ultimate style avatar for user Exponentially Radical
      In laminar flow, the fluid moves quickly near the center of the tube but roughly rests near the walls; this develops a parabolic velocity profile (parabola like x^2 or a quadratic function for the un-initiated). This parabolic velocity profile means that velocity, a symptom of an opposite of resistance, grows with the square of radius. As you mentioned, area also grows with the square of radius. The area (growing with the square of radius) multiplied by the velocity (growing with the square of radius) makes volumetric flow rate grow with r^4.

      --- the rest just complicates things more ---
      When we say that resistance is 16 times as great for the small tube, this means that 16 small tubes would allow the same amount of airflow as one large tube. Clearly, cross-sectional area plays a less obvious role (unintended pun). While Wikipedia shows the integral leading to the higher power of radius, I suspect that you will be most satisfied with the intuition of the parabolic velocity profile.
      Conductivity is flow divided by pressure. Resistance is the reciprocal of conductivity, therefore pressure divided by flow.

      Conductivity growing with r^4 means resistance grows proportional to the reciprocal of r^4
      (44 votes)
  • piceratops ultimate style avatar for user ∫∫ Greg Boyle  dG dB
    @ Besides taking medications, how do we get the arteries to loosen up and vaso-dilate more naturally?
    (14 votes)
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    • blobby green style avatar for user Dave Speener
      By increasing the water intake.

      When you become dehydrated, 66% of the water loss is from the cells, 26% is from the fluid held outside the cells and 8% is from the blood. This may not sound like a significant amount, but when you consider that the blood is made up of 94% water, it's enough to make a difference.

      The reason the arteries constrict is to take up the slack when the blood volume decreases. Failing a capacity adjustment to the “water volume” by the blood vessels, gases would separate from the blood and fill the space, causing “gas locks”.
      (6 votes)
  • piceratops ultimate style avatar for user D.J. Smith
    what's viscosity? (This might sound daft, but I'm not a native English speaker, and translating a language's specific vocabulary to another is hard ...)
    (12 votes)
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  • hopper cool style avatar for user !*Kh@n-U$er*! @k@-(Christie)
    On the Arterioles, is their a specific name for the smooth muscles surrounding it?
    (10 votes)
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  • starky ultimate style avatar for user Mal ❤
    I don't understand. Since the veins are very large, why do they have lower pressure than asteries, which are smaller?
    (4 votes)
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    • piceratops ultimate style avatar for user Ivan Occam
      In general, any pipe with a larger diameter will have a lower pressure. Pressure is the ratio of force divided by the area it is distributed over. So the same force will exert higher pressure in a smaller tube than a big one.

      As far as the human body goes, veins are also less pressurized because they have very high capacitance (the ability to stretch), and they're far away from the heart. The arteries on the other hand, have to handle all the high pressure blood being pumped out of the heart. Arteries very close to the heart are very elastic so that they can stretch to accomodate this high pressure without tearing.
      (7 votes)
  • leafers tree style avatar for user Jane Riverside
    What is the cause of Vaso Constriction?
    (3 votes)
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    • leaf green style avatar for user Nahn
      Overall, vasoconstriction happens when the smooth muscle cells that are found mostly within the middle layer of your artery get a signal to contract, which "Constricts" or narrows the opening of the vessel.
      There are a lot of signals that can tell blood vessels to contract. If they are stretched they can immediately contract as a reflex. They will also contract in response to certain substances produced by the body (such as epinephrine or thromboxane). Or if your body is exposed to cold temperatures, your nervous system can send signals to the blood vessels that are the closest to the skin surface and tell them to contract so that less blood flows through them and you do not loose too much body heat through your skin.
      (6 votes)
  • starky ultimate style avatar for user james
    why do our blood vessels need to contract and relax?
    (4 votes)
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  • duskpin ultimate style avatar for user oskargonzalez
    I'm not understanding nor appreciating the relationship between pressure and resistance. As fluid enters a restriction or smaller radius in a tube, resistance increases (Poiseuille) yet pressure decreases & velocity increases (Bernoulli).

    Increased resistance, I'm picturing crammed molecules which is the same thing as increased pressure? Help.
    (4 votes)
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  • leaf green style avatar for user christie2taylor
    How do you reconcile
    1) the extra resistance/pressure/work in a smaller tube according to Poiseuille's Law
    with 2) Bernoulli's equation, which says that pressure decreases as the tube gets smaller, and the fluid actually flows more readily?
    (4 votes)
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  • spunky sam blue style avatar for user Kashika
    I doubt that there can also be reason why it is harder to push air through the smaller pipe since the area of the pipe is small and pressure is inversely proportional to area so it will end up with more pressure that is why it is harder to blow air out of it. Am I right? If not than please explain.
    (3 votes)
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    • purple pi teal style avatar for user Joana
      You are absolutely right! Actually, that is what the formula says!The formula is R=P*L/A, being L the length, p the viscosity and A the area. Since in this situation we're picturing a tube, the area is pi*r^4, so both you and Rishi are right.
      (2 votes)

Video transcript

So let's say you're walking down the street, and I'm going to draw you here, and you decide to do a little experiment. You take a deep breath as you're walking, and you decide to blow out through a cardboard tube, something like that. And let's say it's like a toilet paper roll. And so you blow out through it, and here in this toilet paper roll the length is, let's call l, and here there is a radius, and I'll call it r. And we know that toilet paper roll is, let's say, about 2 centimeters radius. And I may not be exactly right, but let's just assume that. And you do this and you find that it's so easy to do. Very easy to take one breath of air and blow it out through a cardboard tube. And so you decide to do it slightly differently. So you do it again, and now you do it slightly differently. Now, you go ahead and take an equal-sized breath, but instead of a tube, you choose a straw. And that straw is obviously much skinnier. And you try and you find that it's actually really hard to blow air through that straw. Not as easy as it was before. And so you find that it's much, much more difficult, and you want to figure out why. And you know the tube was l and this straw is the exact same length, l, and the radius now is smaller. So instead of r, let's call this r prime. And instead of 2 centimeters, this one is about, let's say, 1 centimeter. It's a pretty big straw still, but let's just assume that for the moment. So you want to figure out why in the world is it harder with a straw? And this question posed slightly differently was asked actually a long time ago by a French gentleman by the name of Dr. Jean Louis Marie Poiseuille. And I'm actually probably mispronouncing that a little bit so I apologize to any of Dr. Poiseuille's relatives. But this is a Frenchman, and I'm going to spell out his name for you. He actually lived in the 1800s, was born actually in the 1700s, but lived in the 1800s and-- this is a u-- and in 1840s, he put together a set of equations that helps us answer that question that I just asked, which is about why was it more difficult in one situation to the next. So he said if you have a tube and you're trying to get a fluid through that tube. So in this case not air, but a fluid. But you'll see that a lot of the math is very similar. He said if you know the length of the tube-- and let's call it l just as we did before-- and if you know the viscosity, and viscosity he calls eta. This is viscosity of the fluid. If you know these things, and lastly, if you know the radius then you can actually calculate the resistance. And he said the resistance and I'll just call that R from now, big R-- not to confuse you with little r, which is radius-- equals 8 times the length times the viscosity divided by the number pi times the radius to the fourth power. Now, that might look confusing, but look at this. We actually have a lot of these values. We know the length of the tube. We can probably figure out the viscosity, and all we need to do is measure the radius, and we have the resistance. So it's a pretty powerful formula, and we can actually use this to understand what happened earlier. So in this earlier example-- I'm going to go back to this now-- let's take that resistance formula. So big R equals and we said 8 times l times it was the resistance over pi times r to the fourth. And I'm just going to go ahead and replace this with resistance is proportional, therefore, to 1 over r to the fourth. And you can see why that's the case because all this other stuff can be figured out in this example. And you can see that there's a relationship between these two things-- let me draw them-- R and little r. And the relationship is stated here, right? So you have as the radius gets very, very, very big, the resistance is going to get very, very small. And, in fact, it's going to happen very quickly because you're raising the radius to the fourth power. Now, let me take this one step further. Let's look at the other side now. So over here we have a situation where we said we have, again, resistance equals 8 times the length times viscosity divided by pi. So far, it should look the same, obviously, right? And here's the big change. So instead of r, I'm going to say r prime to the fourth power. Now, what's the relationship between the two things? So we said that if r is 2 centimeters, r prime is 1 centimeter. That means that r prime equals r divided by 2. And, of course, I made up these numbers so that relationship is just for this example. And so if r prime equals r over 2, I'm going to replace that in my equation. So just as before, I'm just going to say R is now proportional to 1 over r prime to the fourth power, which means that R is proportional to-- I'll just keep writing it out-- 1 over r over 2 to the fourth power, which means R is proportional to 1 over r to the fourth over 16. Because 2 to the fourth power is 16, right? And it's in the denominator here. And so if we flip it up to the top, we see something really, really cool, which is that it's 16 over r to the fourth. So in other words, compare the first example where you had R proportional to 1 over r to the fourth, and now you have R is proportional to 16 over r to the fourth. That means that it's harder because the resistance is-- let me just write that out again-- 16 times greater. That's remarkable. You just dropped the radius a little bit. You went from 2 centimeters to 1 centimeter, and the resistance went up 16 times, and that's why it was so hard. So now let's apply this to blood vessels. So if you now understand the idea that you have blood vessels, and they're basically like tubes and you can probably see where this analogy is going, there are parts of the blood vessels that are called arterioles. So that's a part of the circulatory system. And these arterioles have a very interesting property, and that is that if you look at them closely, they're covered with smooth muscle. So all these bands of smooth muscle wrap around this arterial. And what it does is that when the smooth muscle is relaxed, let's say it's very relaxed, then you get something like this. And if it's, let's say, squeezing down or tightening, let's say constricting, then you get something like this. And you can probably guess what I'm about to draw, something like that. And I haven't drawn the smooth muscle here, but you can still guess it's basically like this. These are just wide, open, relaxed, completely at ease, and these are tight, very, very tight. And that's why they're bringing the diameter and, therefore, the radius of the vessel down. So this radius compared to this radius is just like our cardboard tube versus our straw. And so when it's relaxed, we call this vaso-- vaso meaning vessel-- dilation. And when it's constricted, we call it, same thing vaso-- for vessel-- constriction. And the reason that we want to make this distinction is that we know, based on this example now, that when you have vaso-dilation what that means for the blood is that you're going to have very low resistance so blood can flow through with very little resistance. And when you have vaso-constriction, now you can see why. That means you're going to have high resistance. And so we owe a lot of this understanding and thinking to Dr. Poiseuille from the 1840s.