 We're now going to take a look at what the difference between streamlines means. And if you recall from the earlier segment, we're dealing with two-dimensional incompressible flow. And we said that with the stream function, we talked about the definition of v cross dr, where dr was a vector element going in the direction of the stream function. And we said that if dr was on a streamline and you cross it with the velocity vector, that would have to equal zero. And consequently, what that means is that the velocity vector is always tangential to a streamline or a line of constant stream function. So there can be no flow across a streamline. So what are the implications of that? Let's try out a schematic. So there we have a schematic of a number of streamlines or stream functions. So we have stream function of psi one, stream function of psi two, and a stream function of psi three. And what we're going to do is we're going to take a look at the flow rate across a number of those lines. But from what we said that the flow always needs to go tangential to the streamline, what that means is that between any of these lines, the flow rate needs to be constant. And so we can write, given that each of those lines is going between stream functions, the flow rate between any of those lines needs to be the same or along any of those lines. And so what we'll be doing is we'll be taking a look at the analysis of that. We'll begin by evaluating the flow rate along AB. So across AB, we can write that the volumetric flow rate is equal to the U component of velocity. Now, if we look back at AB, you might have a V component of velocity, but it's going perpendicular to the flow direction or the mass flux or volumetric flow rate. So only the component going to the right is going to contribute. And so that is how we evaluate the volumetric flow rate there. And what I can do, I can make a substitution for the stream function. And so that gives us an equation. Now, along AB, the other thing that we know is that the value of X is equal to a constant along line AB. And so if we look at the total derivative of psi, and we saw this in the last segment, we can express it this way in terms of partials. And if X is a constant, that means that that term is not contributing anything. And the other thing to notice is that this term here is actually the term that is in our integral up there. So we're going to make a substitution. And what we then obtain is QAB. It's the total derivative of psi. And when you integrate that, you get psi at 2 minus psi at 1. So what that shows is the volumetric flow rate is just the difference between the two stream function values. And now what we're going to do, we're going to do this for another one of the lines. And we'll look at the line BC. So we'll now work on this one. So let's take a look at the volumetric flow rate between those along that line. And in this case, the flux across that line is going to be perpendicular to it. So we can express it in the V component of velocity and making a substitution for the value of our stream function as we have that. Now along BC, looking back at that line, we see just like before, along BC, y is equal to a constant. So we're going to use that. And what that means, looking at the total derivative of psi. Now if y is a constant, then the second term here cancels out. And just like before, this here is in there, except it's the negative of it. So we're going to make a substitution. And when we integrate that, we get, oops, sorry about that, we get psi 2 minus psi 1. And if we look back, that is exactly what we got between A and B. Therefore, we can write QAB equals QBC. And then that is equal to the volume flow rate per unit depth. So that is one of the principles of the stream function. If you take the difference from one stream line to another, one stream function line to another, you get the volume metric flow rate per unit depth. Now we've been talking about Cartesian coordinates. If you look at 2D incompressible flow in the r theta plane, so this would be for cylindrical or polar coordinates, continuity for that can be expressed in this manner. And in terms of stream function, we can write this. So that would be the stream function definition if you were dealing with polar coordinates. But what this segment has told us is that the difference between stream functions is equal to the volume metric flow rate. So we'll continue on in the next segment looking at an example where we will solve for the stream function given a velocity field.