 In this lecture segment what we're going to do we're going to take a look at the stream function and in the remaining segments of this lecture we'll also take a look at the potential function but we'll begin with the stream function for two-dimensional incompressible flow. Okay so if you recall earlier on in the course what we did is we looked at the equation of a streamline. It turns out that you can mathematically describe streamlines and you use a mathematical function called the stream function for that so that's what we're going to be taking a look at. So our stream function we give it the symbol psi and that's very common in fluid mechanics and we're looking at a two-dimensional incompressible stream function for this lecture segment and the definition of the stream function itself what we do we break down the u and the v component so the u component of velocity is expressed as being the partial derivative of the stream function with respect to y and the v component is the partial derivative of the stream function with respect to x but it's the negative of that so it's minus partial psi partial x and here psi is our stream function and it is valid for 2d flow only. So the place where we're going to begin we're going to begin with the continuity equation and we'll check and see if the stream function satisfies that so if you recall the continuity equation we had partial rho partial t plus del dot rho v and here what we're dealing with is we're dealing with a two-dimensional incompressible and given that it's incompressible the density is not going to change and then that will also disappear from there so what we're left with for 2d incompressible is del dot v equals 0 so I'll expand that out and what we have is partial u partial x plus partial v partial y equals 0 so that's the continuity equation now to see if the stream function satisfies that what I'm going to do is take the relations up here that we said have defined the velocity components using the stream function and we're going to try to plug it into this equation if all works well it will satisfy it so let's see what happens so when you plug in for the u let me just write that again oops we had partial u partial x plus partial v partial y equals 0 and we said u was defined as partial psi partial y and v was minus partial psi partial x so we substitute those in and we see we get this for continuity and we get 0 equals 0 therefore the stream function satisfies continuity which is good the next thing we're going to do we're going to take a look at what does the stream function mean along a stream line so let's look back at what we said a stream line was and so if you recall what we said that a stream line was was a line that is always tangential to the local velocity vector and we also had the thing that we referred to as being the equation of a streamline so what I'm going to do is I am going to assume that we have some differential element it's a vector of length and we're going to say and it's going to be dr and we say if it's an element tangent to the streamline if that is the case then we know we can use the cross product and if we do v cross dr that would mean that the dr needs to be tangential and and so if it is tangential and we're crossing it with the velocity vector v that would imply that that would equal to 0 and so let's see what happens when we plug in values so I have that and now we're doing the cross product and remember the cross product if you go back and review the vector operators it results in a vector that is perpendicular to the two vectors being crossed so if we have an i and a j component and that will take us into the k so we have k and then evaluating the cross product we get u dy minus v dx and we're saying that that along a streamline is going to be equal to zero assuming that this dr is along the streamline therefore what we can write so a vector that is going in the direction of the streamline this would be true so what we're going to do we're going to plug in the stream function and see if it satisfies that so plugging in the values we're seeing that has to be equal to zero but when we look at this what is this that this is nothing but the total derivative of the stream function so d psi is going to be equal to so what this is telling us is that along a streamline if we're moving along a vector that is tangential to the velocity field at every location d psi is equal to zero so what that tells us is that along a streamline the derivative of the stream function is equal to zero or psi is equal to a constant so that tells us that this function that we've come up with the stream function if you're following a streamline it is not going to change it'll be fixed along a streamline so it's constant along a streamline so what does that mean that means if you have this being your streamline and psi will be equal to constant here and if you keep following that streamline psi is the same constant now it doesn't say anything about going to a new streamline and it will turn out we'll see that that would be psi is equal to constant i'll say constant one which is different psi equals constant one so it'd be a different constant as you go to a different streamline but anyways those are two properties of the stream function it satisfies continuity and as you move along a streamline the stream function does not change