 Okay, in this lecture we've spent quite a bit of time talking about stream functions and potential functions and what we're now going to do is just kind of review and bring some perspective to all of this. So to begin with, for the stream function we said the value was psi. So when we looked at the stream function psi we used continuity to derive it and we showed that it actually satisfies continuity. We also found that the stream function was always parallel to the velocity, the velocity vector. We said that it was valid only for 2D flow and the last thing we did is we looked at the change from one stream function or stream function value to the next one and we said that was equal to the volumetric flow rate between those two stream lines. So that was the stream function and the stream line. We also looked at velocity potential and that was phi. So when we look at the velocity potential we found that that existed only for irrotational flow, irrotational flow being del cross v equals zero. We found that it was valid in three dimensions. We also showed that it was perpendicular to the stream function and the way that we determined the velocity potential we got that from an analytic relationship of u and v and the same technique as we used for solving for the stream function. So those are the stream functions, stream lines and velocity potential. Now what do we do with these? Well, let's just go on here. So if it turns out that our flow is incompressible and inviscid that is what we would call an ideal flow. And this is an area that was studied many, many years ago and it was studied because it was simplified. There were simplifications that occurred and the mathematicians and scientists could study fluid flow in ways that when you have viscosity and it makes it much more complex. And so looking at these two, if we have incompressible and irrotational, we know through continuity that the incompressible comes out as being the following, del dot v is equal to zero. And if our flow is irrotational, we know that del cross v is equal to zero. And through a vector identity, we could then express the velocity field in terms of the scalar potential function. So what we're going to do, we're going to combine those two together. And so what we can then write is del dot v, that's through continuity, is del dot and I'm going to put in here the grad phi of our potential function. When you do that, you get del squared phi and that has to be equal to zero through continuity. Now this equation, del squared phi, is one that has been studied quite often in mathematical physics and is a very, very well known equation and that is Laplace's equation. And the significance of this, I'll expand out the Laplace equation in three dimensions. Oops, sorry, I was thinking three and that's where the three came out, I apologize, that should be squared. So that is the Laplace equation. It is a partial differential equation. So del squared phi equals zero. It is a linear partial differential equation. And what is important about that is that you can superimpose solutions. The Navier-Stokes is a nonlinear partial differential equation, so you cannot do that with a Navier-Stokes. But given that the Laplace equation is linear, you can superimpose solutions. And so this equation, Laplace's equation forms the basis of an area called potential flow theory. And sometimes it's also called ideal aerodynamics. But it was studied quite extensively and you can do a lot of things with it. But what you can do, there are different solutions to Laplace's equation, such as a source flow where you have fluid coming out. You can have sink flows where you have fluid coming in and we can mathematically come up with a potential function for all of these. There is vortex flows where you have vorticity or circulation in the fluid. There are doublets, which is a source and a sink sitting right next to each other. And you get all these very interesting patterns that come out. And we also have freestreams, so we can have a freestream like that. So by themselves, that would be a source. This is a sink. That's a vortex. That is a doublet. And this last one here is a freestream. Now it might not look all that exciting just by looking at these. But through Laplace's equation, you can combine all of these together and so you can start to construct flow fields. For example, you could construct the flow over a cylinder with circulation. And so you'll see there will be a stagnation point here and here and then you'll get all the streamlines coming around, actually goofing this up because it has to be symmetric. It cannot be anti-symmetric. But anyways, you get this type of thing and you can calculate a lot of very sophisticated flow fields using very simple math. You can even solve for the flow over an airfoil. They did these transformations that they would go from a flat plate and they would look at the flow over a flat plate and you'd have streamlines doing something like this and there would be a stagnation point there. But they would transform from what they call the circle plane where you would have the same sort of flow field. And by going through a Jukowski transformation or different transformations, they could go between the circle plane and the physical plane. And anyways, that was done in ideal aerodynamics. I'm sure you can look it up and find information on it. It has also led to things such as paneling codes. And paneling codes are numerical methods that enable you to compute the pressure distribution about bodies. So you can have an airfoil like that. You break it up into a bunch of little panels. Each of the panels has different singularities on it, a source is saying you can have vortex panels. But with that, you can then model the flow field. So provided that you do not have separated flow, that you do not have large, you're not modeling what's happening in the boundary layer, but you're modeling what they call potential flow. And so you get all of those things. It cannot be rotational flow and consequently you're neglecting viscosity here, but it's pretty good provided you don't have separated flow. So anyways, that's kind of a little bit of a rant about where you can go with this. We're not going to do any more with it in this course just because this could be a course onto itself and you could spend a lot of time looking at these things. But I just wanted you to realize that we've been looking at stream functions and velocity potential. And for new students and fluid mechanics, it can kind of be a little bit on the confusing side. You're wondering why are we even doing this? This is where it goes, but it takes a little bit of time to get there. So that's where we will conclude for differential analysis of fluids.