Now that we have an idea of continuity that the flow rate has to be conserved from one place
to another, let’s look into Bernoulli’s principle before we go to our last section here.
Basically, Bernoulli’s principle is simply a way of saying that energy is conserved.
So, if I have a fluid flowing through this pipe as you see it here but then it lifts up to some
other area of the pipe that is at a higher elevation, what we’ve done is given the fluid a higher
gravitational potential energy. So, you might expect that if energy is going to be conserved,
there should be a difference in how the flow is happening in the wider part of the tube
that’s lowered down and in the higher part of the tube which is smaller but also raised
to a different level. First of all, we can just start cataloging all the different kinds of energy
that we can discuss with regard to a fluid. It could have an internal energy, just energy
of the different particles and molecules moving inside of it from a pressure that it has.
It could also have a kinetic energy because it has some non-zero velocity. As we discussed,
the kinetic energy is ½ mass times velocity squared. So, each one of these sections of fluid
will have a kinetic energy of ½ mv squared. They can also have a gravitational potential energy.
We discussed that gravitational potential energy is equal to mgh. So, if each of these sections of fluid
has its own height above some coordinate system’s origin, each section here has its own
gravitational potential energy as well. So, here's a way of writing all the internal energies.
The trick here, and the thing that makes this look more like Bernoulli's principle rather than
something else, is that we can divide this entire equation by the volume. In that case,
we have energy divided by volume which is as we talked about, a pressure, energy per unit volume,
joules per cubic meter. So, that’s the internal pressure, the internal energy just coming from
the pressure of the fluid. Instead of mass, we now have mass divided by volume which is density.
So, ½ ρv squared instead of ½ mv squared. Similarly, in the mgh term, we now have ρgh
because we’ve instead got mass per unit volume. The fact that these are conserved,
that the energy is conserved simply means that this quantity, internal pressure plus kinetic energy
plus ρgh from the potential energy has to always be conserved. So, Bernoulli’s equation
is simply saying that two parts of your system, say one and two, you have this entire quantity
written out, will be conserved from one part to the other. Again, this is just a different way
of writing out the conservation of energy. That’s where this equation comes from.