Bernoulli's Principle

Let's talk about Bernoulli's Principle, Bernoulli's Principle is one of the most important equations that governs the study of fluids that are flowing along in this nice ideal way that's not turbulent, we're not talking about turbulence. We're just talking about nice uniform general flows of fluids. Now this principle was developed by Daniel Bernoulli in 1738 it turns out that it'll replace Pascal's principle as well as the fluid pressure formula that we have when the fluid is flowing. And this is going to be much, much more general. So it'll actually turn out that both Pascal's principle and the fluid pressure would just follow from Bernoulli's Principle once we use it properly. Alright so let's go ahead and look at a diagram of a fluid that's flowing through a pipe. It will turn out that it doesn't actually have to be flowing through a pipe but we're going to kind of imagine putting a pipe around it just to derive the equation.

Alright, so we've got down here a small cross sectional area, the part of fluid over here is at a height h1 above the ground, we have the small cross sectional area a1 and we have the speed v1 and we're going to look after a time delta t has gone by so this part of the fluid will have moved a distance v1 times delta t. Over here at the top, we've got a much larger cross sectional area a2, we have another speed v2 and this fluid is at height h2 above the ground. So now what we're going to do is we're going to apply conservation of energy to this situation. Alright, all the fluid in the middle has just stayed the same, it hasn't even changed it's just doing whatever it was doing before so it doesn't matter. The only effect of this timed, delta t that's gone by was to move a piece of the fluid from over here to over here. Alright so how much work was done on this mass as this piece of the fluid move? Well the net work was equal to the force on the left because that's pushing on the right in the positive direction and the direction that the fluids is moving in.

Well what is that force? Well it's the pressure times the area because that's what force always is, pressure times area. So that work isn't just equal to force, work is equal to force times displacement. So the force is pressure times area times how far it was displaced and then up at the top we've also got work that was done but here the force was pushing the other way. The fluid moved that way but the force was pushing that way, so that means that it's a negative sign, so it's going to be minus pressure, area and then displacement again. Alright now that net work as a result of the work energy theorem will equal the change in total energy of the fluid. Okay, well that's going gravitational potential energy change so mgh2 minus mgh1. Notice that the masses are the same because even if the fluid is compressible, even if it can squish down where has the mass gone? The mass has to go somewhere, so all the mass that was in the bottom part got to go to the top part. So those masses have to be the same. So we've got mgh2 minus mgh1 plus and then we've got the change in kinetic energy.

One half mv2 squared minus one half mv1 squared, alright now the standard thing to do here is to divide by the mass. Mass is equal to density times volume alright, now the volume maybe different and so might the density in each place but it's always equal to density times area times speed times delta t. And notice what we've got, area times speed, times delta t, area times speed times delta t. So when we divide by the mass it'll turn out that the density comes underneath here and of course the mass just cancels everywhere else which was the reason why we chose to divide by in our first place. Right so if we rearrange that, we'll end up with the following expression. Pressure divided by density plus acceleration due to gravity times height plus one half speed squared has to be constant. So this kind of plays a role of energy conservation but inside of a fluid and that's actually the nice about Bernoulli's Principle is that it allows us to think about the fluid in terms of energy conservation.

But this is a little bit better if the fluid is incompressible because in that case the density is constant and we can just multiply by it and then we can actually look at it directly. So I've got rho gh, that's like mass times g times h so this is gravitational potential energy, one half density speed squared. Well jeez that looks just like kinetic energy to me at least, so what about that guy? The pressure it'll turn out has to do with fluid potential energy and it's kind of a new thing. And it leads to a lot of the great properties of Bernoulli's Principle because it tells us that in order to exert pressure, a gas has to have enough energy to do so not just a gas, any fluid. So if a gas is moving then it has to waste some energy on kinetic and that means it can't exert as much pressure. So moving gasses and this is often what people talk about when they say Bernoulli's Principle moving fluids exert smaller pressures than fluids at rest.

Now we can see this directly just by driving down the car, I mean I've got long hair I don't know if you've got long hair but I'm certain you know somebody who's got long hair, roll down the road and you'll see that, that hair goes right out the window why? Well the fluid, the air inside of the car is rest whereas with respect to that air at rest inside the car, the air outside the car is moving. The air inside the car exerts a greater pressure than the air outside the car and that's what takes the hair out. Okay now of course that's not usually what people talk about Bernoulli's Principle with respect to, usually they're talking about airplanes so let's just look at this real quickly. If we have an airplane wing that looks like that and we push it through the air in this direction, the air is going to form streamlines around it like that, so it's moving on the top. But on the bottom we have this little piece of stagnant air, so the moving air exerts a smaller pressure than the stagnant air underneath and that's what gives us this lift. And that's why a lot of times people say that Bernoulli's Principle is the one that allows us to fly. And that's Bernoulli's Principle.