Now that we’ve discussed the circuit law as it’s called for fluids as well as Poiseuille's law for resistance,
let's go to the last part which will be the use of fluid laws because there are some misconceptions
that are very common when applying, especially the conservation principles with fluid laws
that we should go over. Before we go into that, here are some of the things that we’ll be using
in this example which is a basic overview of what we’ve introduced so far. You would probably
like to review this, maybe pause or use this as a resource and make sure you understand
all of these variables especially what these variables mean physically if you actually have an image
of your system in your head. What we’ve done is we’ve introduced a fluid flow rate which is the
cross sectional area of a vessel times its velocity. We introduced the circuit law which has to do
with pressure and relating it to the flow rate with the resistance. We then introduced Poiseuille's law
which gives us a particular expression for the resistance to plug into the circuit law which we did.
We talked also about the conservation of flow in a given system since the mass has to be conserved
and we can have gaps or overlaps in our system as well as talking about Bernoulli’s principle
and then the resulting Venturi effect when you have an increased velocity. So make sure
you’re familiar with all of these. But we’re going to go into some examples now to see how to properly apply
some of these conservation principles. The basic mistake that is very easy to make
when you’re talking about a conservation law is that the conservation laws we’ve talked about so far,
as you can see here, are referring to different parts of a given system. Meaning that we look at one part
of a system, maybe the wider part on the left and then we talk about a different part of the same
given system, maybe the part on the right which might be a slightly thinner vessel. It might be elevated
relative to the original. Then we talked about conservation using Bernoulli’s law
or the flow continuity equation. What we’re going to have to be careful with is what if we’re talking about
a question which asks us about two different systems? Maybe an original system that we then changed
so that it is a new sort of system. Then we have to be very careful applying our conservation laws.
So, we’ll see that here. Here’s a first example. Suppose the pressure in a system is doubled
because we add some sort of a pump. Again, when we say the word pump often in terms
of your cardiovascular system, we would be referring to your heart which is the organ
that’s applying that pressure. The question is if we double that pressure in a given system,
what would happen to the velocity if everything else on the system remain the same?
A very common or easy mistake to make here might be to think that the velocity has to decrease
because the pressure increased. When we were talking about Bernoulli’s principle, we derived
an equation like this one which is one where all of your vessels are at the same height.
Then we talked about the Venturi effect which we said that when you had a higher velocity,
you had a decreased pressure. So in this problem, you might think, "Well, the pressure increased
so the velocity has to decrease by the same principle." But in fact, that wouldn't be the case
because this is not a single system where we’re talking about two parts of that system.
Instead, it’s two different systems. A first one with some pressure and then we changed our system
to be a new system where we've doubled the pressure. So, we can't simply compare two parts
of a given system which we might be tempted to do here. A correct answer would instead be
to say that the velocity increased because the resistance wasn't changed while you did
increase the pressure. So, using our circuit law for fluids that we see here, we know that the flow,
the flow rate through the system is proportional to the change in the pressure, the pressure difference
that you're applying to your system divided by the resistance of that system. Looking at this equation,
if the resistance is not changing, if we have the same system in other words but we double the pressure,
we can see that the flow rate, the area times the velocity of your system has to double as well.
So in this case, the velocity would in fact increase and not decrease. A second example
might ask you that for example, due to plaque, the radius of a particular blood vessel in your system
is decreased by half. So, what would happen to the pressure in that particular vessel if the overall
cardiovascular system is negligibly affected. So, there’s a lot to think about in this problem
a lot to internalize. So, a tempting thing to do might be to just think, “Well, the pressure increased
but we know that the flow rate has to be constant.” So, just using the circuit law here,
the pressure has to increase to keep the flow rate constant rather. This would be incorrect
because again in this system, we're not talking about two parts of the same system.
We’re instead talking about a particular system that then changes to a different system
which has a vessel of half the radius that it had before. A correct answer would instead be to say
that the pressure has to decrease because the velocity increased. This is just the Venturi effect.
Briefly, I should mention why we’re allowed to use the conservation law from Bernoulli
rather than the conservation of flow even though I’ve said that we’re talking about a different system.
So again, you should think about this very, very physically. You have this entire cardiovascular system.
You’re applying some pressure to it and the blood is flowing. But then, one part of that system
has an increased resistance because the radius, the size of the vessel that the blood could go through
decreased. So if that happens, we might think well, if we’ve increased the resistance
but we have the same pressure being applied to the system from the heart for example,
we might think that well, the entire flow rate through that system has to decrease.
That’s where we'd like to use the last part of this problem as its stated. We're told that the entire
cardiovascular system is negligibly affected because one tiny vessel in the entire system
which is quite a big system relative to the size of that vessel is not going to change
the entire system very much. So if that’s the case, we can say that the overall flow rate
through your system is not affected .The overall pressure being applied to your system
is not affected as a whole. So, our other conservation principles would still apply
because we have the same flow rate. So, we wouldn't need to use the circuit law in this case.
So in that case, if we’re not changing the flow rate through the system, in this case we’re still
talking about the same pressure being applied on either side of this blood vessel below here.
We could say that the velocity to that particular part, the part that’s been shrunk will have to increase
to keep that particular flow rate which we just said isn't changing the same. A few more here:
Let’s say that now due to plaques in your entire system, many blood vessels have restricted their radii.
So now, you have many restricted blood vessels. We could ask what happens to the flow rate
in the system before the body has adapted. What we mean by this before the body has adapted
is that if all the radii of many blood vessels have changed, perhaps your heart hasn't started
beating harder or gained strength to push more pressure into your system. In other words,
we’re not going to allow the system to have changed its pressure. Looking at this question,
we might think the flow rate is constant. Therefore, we can just say the flow rate hasn’t changed.
The flow rate is constant. So, if we're asked how is the flow rate changed, it’s very tempting just to say,
well it hasn't changed. We learn this principle of constant flow rate over and over.
Let’s just say no change. But that wouldn't be correct in this problem. It wouldn't be correct
for the same reason that we discussed before which is that instead of describing two parts
of the same system, we are now describing two different systems. So, we have a system before
the radii of all these blood vessels changed and then we have the system after this change.
So, a better answer here would be that the flow rate decreases because all the resistance
has increased. The applied pressure applied to your system again, maybe from your heart
hasn't changed because we haven't allowed this system, in this case the cardiovascular system
to adapt and to allow your heart to really apply a lot more pressure to get that flow rate back up
after all these plaques have increased the resistance in your system. So again, be careful
when you're comparing a system in different parts of their given system. We're comparing
two changed systems one to the other. So one final example: What if we took this exact same situation
but now we say that the system has been allowed to adapt. So due to plaque one more time,
we have the radii of many blood vessels decreasing which does affect the entire system.
But now, we ask what happens to the pressure of the system which has now adapted to increase
the blood flow back to the normal levels that your body would need, a particular blood flow
to keep all the different organs in your system going with oxygen. Again, an incorrect answer here
would be to say that well, the velocity has increased and so the pressure has decreased.
It would be very tempting in other words to look at these two pictures here and say well,
the radius of this vessel has decreased. Therefore, the flow has to go much faster to this vessel.
Therefore, the velocity is increased. We knew it by the Venturi effect that the pressure decreases.
Again, this would be incorrect for the same reasons that we've discussed. That you're not comparing
two parts of the same system. In other words, we're not talking about the part to the left
of this constricted area and then the part in the constricted area. We're instead talking about
an entire system where the radii of many parts of that system have decreased and so the overall
resistance in the system has increased. For this reason, a better answer would be that the pressure
has to increase. Your heart has to apply more pressure to deliver the same flow rate.
So looking at our circuit law that we have written here, supposing that the resistance went up
and you want to keep the same flow rate, you’re going to have to increase the pressure,
the pressure difference applied across your system in order to get the blood to flow
through that system. This is exactly what happens if you do have a lot of plaques
in your blood vessels. Your heart has to adapt. It has to start pumping harder, applying more pressure
to that system. In that case, you can get a higher blood pressure. It’s very important
to understand this and to be able to get questions like this right. Because you certainly wouldn't want
to say that somebody who has a lot of plaque in their system would have a decreased pressure
in their system if their heart is still trying to deliver the same flow rate. So, with these examples,
we have a few concrete cases of how you could apply some of the conservation laws
that we’ve discussed as well as the circuit law and those few examples of Poiseuille's law as well.
This wraps up our discussion on more complicated, more applied hydrodynamic systems.
Thanks for listening.