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### Course: MCAT > Unit 9

Lesson 7: Fluids in motion- Fluids in motion questions
- Volume flow rate and equation of continuity
- Bernoulli's equation derivation part 1
- Bernoulli's equation derivation part 2
- Finding fluid speed exiting hole
- More on finding fluid speed from hole
- Finding flow rate from Bernoulli's equation
- Viscosity and Poiseuille flow
- Turbulence at high velocities and Reynold's number
- Surface Tension and Adhesion
- Venturi effect and Pitot tubes
- Two circulations in the body
- Arteries vs. veins - what's the difference?
- Resistance in a tube
- Putting it all together: Pressure, flow, and resistance

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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.

## Want to join the conversation?

- 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)
- 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)

- @8:24Besides taking medications, how do we get the arteries to loosen up and vaso-dilate more naturally?(14 votes)
- 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)

- 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)
- Viscosity can be thought of as the "thickness" of a liquid and is a result of the friction between different particles within the fluid. Think about how much easier a fluid like water (with low viscosity) flows than something like ketchup (higher viscosity)(14 votes)

- On the Arterioles, is their a specific name for the smooth muscles surrounding it?(10 votes)
- All arteries have a layer called the "tunica media" which basically means the "middle coat." It is the muscular part of the arteries and is made up of smooth muscle.(11 votes)

- I don't understand. Since the veins are very large, why do they have lower pressure than asteries, which are smaller?(4 votes)
- 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)

- What is the cause of Vaso Constriction?(3 votes)
- 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)

- why do our blood vessels need to contract and relax?(4 votes)
- to regulate blood amount that goes to a specific organ and to adjust our blood pressure(4 votes)

- 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)- I had the same question and asked many people. Unfortunately the only answer I got was compliance and elastance.(3 votes)

- 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) - 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)
- 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.