 We're now going to take a look at the equation that defines a streamline. So in the previous segment what we did is we looked at some different experimental approaches to being able to illustrate where streamlines were within a fluid flow and now what we're going to do we're going to take a look at the mathematical equation of a streamline and then we'll come up with an arbitrary function and look at the streamlines for it. So to begin with let's draw out our coordinate system. So we have the z-axis in the vertical, x and the y in the horizontal and what we're going to do we're going to assume that we have some arbitrary velocity vector. Now in the x direction this velocity vector will have a velocity component little u. In the y direction it will have a velocity component little v and in the z or z component direction it will have a velocity component little w. And what I'm going to do is I'm going to assume first of all I'm going to draw a line to the origin. We're going to draw a little vector, a spatial vector that matches up proportionally with the velocity vector itself and we're going to give it a little differential element let's say it's length dr and then in the three spatial directions that would be dz it would be dx in this direction and it would be dy in the y direction. And so what we're going to do when you look at a streamline remember streamlines have to be tangential to the velocity vector. So geometrically we can come up with the following equation that describes a streamline we can say dx over u equals dy over v equals dz over w is equal to and then in generic sense dr some arbitrary location divided by the magnitude of our velocity vector. So that gives us a geometric representation and equation that we can use to look at a streamline. So what we're going to do next we're going to take a look at an example of solving for the equation of a streamline. So let's assume that we have a two-dimensional steady velocity field and let's assume that the representation for the velocity is given to us in terms of an equation. So this is just some arbitrary velocity field and we're told the u component is x squared minus y squared and the v component of velocity is minus 2xy and we're asked to derive a streamline pattern. So in order to do this what we'll do is we'll go back to the equation that we just looked at and we're going to take that equation and we'll work with it and essentially integrate this to come up with an equation of a streamline. So let's begin by taking our equation of a streamline dx over the u component and the u component was x squared minus y squared is going to be equal to dy over the v component which is minus 2xy and that's equal to dz over zero while that blows up but what this does is it indicates to us one thing and that is that the streamlines are only in the xy plane. So it indicates that the streamlines do not vary with the z dimension. So now that we have this equation what we're going to do we're going to rearrange it and it turns out that we can get an exact differential equation out of this. So if I rearrange this part of the equation here what we get we get an exact differential equation that we're going to equate to df and f is going to be the function that we're looking for. So essentially what we can do we can write out that this is partial f by partial y dy plus partial f partial x dx equals df and so what we find is that this here was our u component or sorry the x squared minus squared yeah with the rearranging it was the u component there but we have this and now what we're going to do is we are going to integrate this each of the components. So to begin with we have partial f by partial y equals x squared minus y squared looking back at our equation that was this right here partial f partial y and so when we integrate that we find that we get f is equal to x squared y minus one third y cubed plus some constant of integration that could be a function of x we don't know but we will say that it could be a function of x or it could just be a constant and for the second part of the equation which is right here we have another equation that we will integrate and so that is df by dx equals 2xy when we integrate that we get f is x squared y plus and then we have some other constant of integration which could be a constant or it could be a function of y so we'll write it that way. Now what we're going to do we're going to equate these two and when we equate these two what we end up with is the following equation so what does this tell us well first of all x squared y cancels out of either side of the equation on the left hand side here we have a function of y plus a function of x and on the right hand side we only have a function of y so what this tells us immediately is that c of x is equal to zero or some constant that is not a function of x and the other thing it tells us is that c of y is equal to minus one-third y cubed so with this we can place these values back into our function that we have up here as well as the other function that we have and we can rewrite the function in the following manner we get x squared y minus one-third y cubed is equal to some constant and this is what turns out to be the equation for the streamline that satisfies the velocity field that we specified at the beginning of the problem here so with that velocity field this is the equation of a streamline for that particular velocity field so what we can do we can take this velocity field and we can or the this field of the stream functions and we can plot it and the way that we plot it is we plot it using a contour plot so what I've done and I'll show you in the next couple of slides is a plot of that function so here we have the function which we said was f so we have that function plotted here and then we have contour lines so these contour lines represent stream lines for that function and if you look at it in color it looks something like this and notice we have x plotted here and y there so the velocity vector would be tangential and moving in these directions here so all of these would be streamlines within our field and the contours are shown there it's an arbitrary function it's kind of an arbitrary field that we're looking at as we go on in the course we'll use techniques that can give us stream functions that are a little bit more realistic and we'll look at that when we do potential flow modeling later on within the course but that is an introduction to the stream function the equation of the stream function and how you can solve for the streamlines for an arbitrary field