Now that we've discussed some basic hydrostatics and hydrodynamics, we’re ready to move to
applied hydrodynamics. As a reminder, with the basic hydrostatics and basic hydrodynamics,
we haven't allowed our fluid that were flowing through a particular system to experience any sort of
resistance or any turbulence or any problems like that. But now, with this applied hydrodynamics,
we’re going to move to a way to think about what happens in a circuit which actually has resistance in it.
What we will do for this applied hydrodynamics talk is first start with something we call
the circuit law for fluids. We’ll see how that’s analogous to the circuit laws that we see when we get to
actual physical circuits with electricity flowing through them. Then we’ll discuss something called
Poiseuille's Law which has to do with resistance and circuits. Then finally, we’ll apply these both
in a few actual examples. But first, again, let’s start with the circuit law for fluids. What we can do
is in a particular system, imagine that we have a resistance where we can define this resistance
by thinking about how much pressure we have to apply to a system to get the fluid to actually
flow through it. In other words, to realize that a resistance in a circuit when the fluid flows through it
will take the energy from the fluid. We can ask how much pressure do we need to apply to this fluid
to get it to flow through a circuit which actually has resistance in it. In other words, if we apply
a particular pressure and you can see here, we maybe have one pressure on one side of the fluid,
a different pressure on the other side which will want to cause the fluid to move through it.
We can ask how much flow do we get for a given amount of applied pressure.
So, we can define a resistance exactly like this. We can ask how much pressure do we apply.
That’s the change in pressure, the ΔP here which asks how different are the pressures on either side
which is what will cause the fluid to flow. Then we can ask how much flow, Q do we get for this
given amount of pressure difference which we tried to apply to get the fluid to flow in the first place.
So, really this equation is answering the question how much pressure per amount of flow
do we have to apply to get that fluid to flow in a circuit which has or in a system which has
some resistance in it. Just by rearranging the equation that we just introduced for, the definition
of the resistance in a circuit, we can arrive at what's called the circuit law for fluids
flowing through some vessels or some system. So, the circuit law looks exactly like this.
It’s a change in pressure is equal to the amount of flow, Q that you get times the resistance
in your circuit. Just by examining this equation, you can sort of see how it’s intuitive.
If you have more resistance, you get less flow for a given amount of pressure.
I highly recommend that you take a look at this equation and really think through it and make sure
that it has some intuitive sense to you as we use this equation because it’s used very often
and it’s very important. The ideas behind it will again come up when we talk about
actual electronic circuits. What we’re doing in this equation is something that you should really
focus on physically. If you think about the pressure, the resistance, as well as the fluid flow,
what you should really understand is that a given physical system, maybe your blood system
has a given amount of pressure applied to it and then also has a given amount of resistance
because of just the physics, the actual materials that are at play. Maybe you have some plaque
in the blood vessels or something like this introducing a resistance. So both the pressure
and the resistance are simple physical realities about a given system. Those won’t usually change.
Certainly not very quickly which means that the fluid flow, the amount of blood flow that you get
in an example with blood flow would be the variable in this equation that adapts to the given pressure
and the given resistance. The question at this point is now that we have an idea of what this pressure is,
we’ve talked a bit about pressure already up to this point and we already have an idea
of what the fluid flow, Q is as the area of a vessel times the velocity of the blood flow, can we somehow
come to an equation or some understanding of what the resistance in a given physical system would be.