Fluids in motion
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Resistance in a tube
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.