That brings us to Poiseuille's law. What Poiseuille's law tells us is that we can write an expression
for the resistance in a circuit if we make some basic assumptions about a circuit. We’ll talk about these
assumptions in a moment. But first, what this equation is telling us is that the resistance in a circuit
will be equal to 8 times μ which is a measure of the viscosity of your fluid. Blood isn’t exactly non-viscous.
It has some viscosity to it. So, if we’re going to talk about a real system, we do have to start
thinking about the viscosity of a fluid times the length of whatever vessel the blood flow
is going through divided by π times the radius of your vessel to the 4th power. From the circuit law,
we can do one more thing which is now that we have a definition for the resistance, we can actually plug in
this expression that we just found for the resistance into the circuit law and rewrite the fluid flow
as a ratio one more time of the pressure divided by the resistance using the resistance
that we just wrote here. As I said, we’ve made some assumptions in writing this resistance.
It’s not going to always apply to every fluid flow. Some of these assumptions are that it is a laminar flow.
So, we’re still not talking about any turbulence or any flow that’s sort of roiling about.
We also have assumed that we have a constant cross sectional cylinder, so a very simple sort of tube
like a blood vessel without any sort of impurities or strange shapes to it. Finally, we haven’t included
in this equation anything about the height for example as we talked about the gravitational
potential energy. So, we don’t have to include anything about that for this equation.
It is a complicated equation. Generally, you won’t be needed to or required to memorize the entire thing
with all of its intricacies. But there are a few important things to remember about the final expression
that we arrived at for the fluid flow. One of which is that the flow, Q will depend on the radius
of your vessel to the 4th power. Secondly, it depends on one over the length of your vessel,
meaning that if you double the length of your vessel, you could have the fluid flow if everything else
in the system stay the same. Finally, the fluid flow will depend proportionally on the amount
of pressure difference that you’re applying to the fluid. So, if your pump in your system for example,
the heart doubled the amount of pressure it was putting into the system and everything else
stayed the same, the amount of fluid flow you get out of that system would be doubled as well.
Now, let’s look at an example of how we could apply Poiseuille's Law. In this example, we’re asked
if we have a blocked airway which now has an effective radius of half its value, how much pressure
do we now have to apply if we still want to get the same flow of air through that blocked airway.
Don’t forget, as we mentioned before that air itself is actually also a fluid following many of the same laws.
So, go ahead and use Poiseuille's law and see if you can solve a question like this one remembering
the proportionality laws that we introduced without necessarily needing to resort to the full equation
with all the variables. So, give that a try. If you tried it, hopefully it looked something like this.
We remember from Poiseuille's law that we had the resistance in a circuit was proportional
to 1 over the radius of that particular vessel, whatever it was to the 4th power. What this will be
if we have the radius will now be 1 over ½ the original radius also the 4th power. This is equal to
1 over 16 times the original radius to the 4th power. Remember, we actually have this as a fraction.
So, we have ½ to the 4th power is 1/16, meaning the resistance is now 16 over r to the 4th
where r would be the original resistance. We can maybe write it like this using this r zero as the original
resistance with ½ of our zero as the new resistance. What we could see now is that this would be 16
times the same 1 over r zero to the 4th or in other words, that our new resistance is equal to 16
times the original resistance. All we have to do at this point is think about what the new pressure
would have to be. We remember our circuit law that the change in pressure across the system
is equal to the flow through that system times the resistance in that system. But now, we have a new
system with a new pressure, a new flow rate, and a new resistance except that in this problem,
what we’re saying is that the flow rate should be the same. So, we would like this to not have changed.
What we can do is simply divide both of these equations by finding the ratio. Let’s do that.
We have the old change in pressure across the system. Let’s actually write it as the new change
in pressure, the new change in pressure across the system as a ratio of the old. Just by using
these equations that we have here, it's now equal to Q times the new resistance divided by the same Q
divided by the old resistance. Again, this question is asking us what happens if these Qs are the same.
Since they’re the same, we can cancel those from the numerator and denominator.
So, this will just be the ratio of the resistances where again, we had an original resistance,
r1 but then we changed it to our new resistance where we see that the new resistance is actually
rather like this, the new resistance, r2 which is proportional to 1 over r to the 4th is now equal to 16
times the old resistance. So, the new resistance is 16 times the old resistance divided by
the old resistance. We can see that the ratio of the pressures will also be 16 since they’re proportional.
So, we can see that the new pressure that your system will have to apply is not just twice
the old pressure when you have the radius of your airway but it’s actually 16 times the original pressure.
So, you can see that when we’re dealing with fluids, we actually have to be quite careful
because sometimes when you’re having the radius of an airway as we did here, the amount of pressure
that you have to apply isn't necessarily simply twice the amount of pressure. When you introduce
resistance into circuits that is slightly more realistic, you can have compounding effects
because your air has to go through a much tighter channel. So, we can see the amount of pressure
you would have to apply which for your airway would come from the force that you’re exerting
with your lungs would actually have to increase quite dramatically even for relatively modest change
in the size of the airway that the air would be going through.