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 that is a radius and I'll call it R and we know that toilet paper roll is let's say about two centimeters radius that 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 up 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 equally an equal sized breath but you 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 our let's call this R prime and instead of 2 centimeters this one's about let's say 1 centimeter it's 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 the straw and this question posed slightly differently was asked actually a long time ago by a gentleman french gentleman by the name of dr. Jean Louie Marie Wausau and I'm actually probably mispronouncing that a little bit so I apologize to any of dr. bosons 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 Ashley in the 1700s but lived in the 1800s and this is au and in the 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 called 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 let's call it l just as we did before and if you know the viscosity and viscosity he calls ADA ETA now this is viscosity of the fluid if you know these things and lastly if you know the radius if you know the radius then you can actually calculate the resistance and he said the resistance resistance and I'll just call that R from now on big R not to confuse it 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 a 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 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 eight times L times was the resistance over pi times R to the fourths and I'm just going to go ahead and replace this with a resistance is proportional therefore to 1 over R to the fourths and you can see why that's the case because all of this other stuffs all this stuff can be figured out in this example and you can see that there's a relationship between these two things let me just 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 rate 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 up 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 4th 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 are 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 I just write that out again 16 times greater that's remarkable you just dropped the radius a little bit you know you went from 2 centimeters to 1 centimeter and the 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 arterioles 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 kind of wrap around this arteriole and what it does is that when the smooth muscle is relaxed let's say it's very relaxed 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 you know 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 it's 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 vasodilation what that means for the blood is that you're going to have very low resistance so blood can flow through with very low resistance and when you have vasoconstriction 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+ off from the 1840s