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# Resistance in a tube

## 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.