 Okay, we're now going to take a look at an example problem involving the stream function. And what we've looked at thus far, we've said that the stream function satisfies the continuity equation. The value of the stream function does not change as we go along a streamline. And the difference between two stream lines is equal to the volumetric flow rate. So what we're now going to do, we're going to come up with an analytic expression for a stream function given an analytic relation for a velocity profile. So let's assume that we're given a velocity field. And a here is going to be a constant. And a equals a constant. And what we're told to do is to find the stream function. So we're going to try to find the stream function given this analytic velocity field that we've been given. Looking back at the definition of the velocity components for a stream function, we said that it was partial, partial y was the u component. And v was minus partial psi partial y. Sorry, partial x. So with that, what we're going to do, we're going to take the two velocity components, we'll begin with the u component and then the v. And we're going to integrate the stream function to try to find out what it is. We're going to get a constant of integration that we'll work through it that way. So beginning that process, we have u partial psi partial y. And then if we look at our velocity profile that was given, that is ax. So with that, what we can do, we can integrate, we get integral of ax dy. And when we integrate that, we get ax y plus some constant of integration. And here I'm going to say that it's a function of x. Because notice we're integrating with respect to y. So really it could be a function of x or it could be a constant. At this point it's unknown. What we're now going to do, we're going to take this that we just solved for for psi. And we're going to plug it into our velocity in the v direction or in the y direction. And so what we need to do, we need to take the derivative of our stream function here with respect to x. We get that. And then that is going to be equal to the v component of velocity that we have up here, which we said was minus ay. So what do we get out of this? Well we see this here matches directly with that. And f of the derivative, there's nothing there. So that's equal to zero. So what we can say is partial f by partial x is equal to zero. Now that would only be the case, that would only be the case if f of x is equal to a constant. So it's not a function of x, it's just some constant. So if we know that, if we know that f of x is equal to a constant, and we know from up here that psi is equal to that, we can combine those two and we then come up with an algebraic expression for our stream function as being axy plus some constant. So what we can do, I'm going to take this and put it into a contour plot and I'll show you the result. And I'll compute this for the case of a equals one, and I'm going to assign that arbitrary constant equal to zero. It doesn't really matter. It's just going to shift the stream lines one way or another. But when we do that, what we get, we will see it on the next slide here. Here we go on the left. We have a color contour map, and on the right we have a contour without the colors. But what you're looking at is the stream function axy plus c, where a is one. So really it's just x, y plus zero. So we're just looking at x, y. And that would be the stream function. And so we're looking at the velocity field. If you recall, we had axi minus ayj. So plugging in our values, that is the velocity field of xi minus yj. And so those are the stream functions that are for that velocity field that we're looking at. So that gives you an example of how to solve for a stream function. You basically just integrate twice and take care of the constant, whatever it might be, as you go through the integration.