 We're now going to take a look at what we call irrotational flows and we looked at vorticity and we looked at rotation rate and we looked at circulation and we saw that the rotation rate could be determined by the curl of velocity and that would imply that your fluid then has rotation but for an irrotational flow the curl of velocity so del cross v has to be equal to zero and so this is an area that we will explore shortly before we get to that I just want to make a comment about what causes rotation within a fluid flow so this is an interesting comment and so let's go through and dissect it a little bit it says a fluid particle moving without rotation will not develop a rotation under the action of a body force or normal surface or pressure force so if we have a chunk of fluid remember we've always talked about this differential element and it's moving along and if the only thing acting on it is pressure and the body force that fluid element will just continue to move along and translate it will not rotate under those two forces alone the only way that it will start to rotate is if it has angular deformation and shear stress so what that means is that there has to be differential shear from the top to the bottom and that will start a rotation of the fluid and and so the presence how do we get that we get that through the relationship between shear stress and that we have that related to a function of velocity and we saw that the viscosity had to be in there so only viscous fluids will start to develop what we call rotational flow and and consequently that is a place where you would not be able to make this irrotational flow assumption now it turns out that for most objects if you're looking at external flow the viscous forces really only start to become really important when you get very close to the body and so as you go further away from the body there we can make the approximation of your rotational flow because viscosity or the viscous forces are not as significant but when you get very near the body we get into the boundary layer and if you're called back in an earlier segment we had the separated flow that's obviously rotational and viscosity is very important there further away from the body we we can make the irrotational flow approximation so so why are we interested in irrotational flow well it turns out that if you can make the irrotational flow assumption it enables you to come up with a thing called a velocity potential and so let's take a look at the velocity potential now and this is valid for three-dimensional irrotational flow streamlines were valid we look for 2d incompressible this is for 3d flows but they need to be irrotational so with the definition of being irrotational that means that the angular rotational rate is zero and del cross v is zero and with this there's a vector identity that if you have a vector that you have del cross v or the cross product of that and they're the curl of that vector if it is equal to zero then through a vector identity we can write that v is equal to the gradient of some scalar field and this is a vector identity and so we're going to take advantage of that and this is actually it's going to be our potential function we saw the stream function earlier now we're going to look at this potential function it's another function that is a function in this case of the three spatial coordinates and we can also have it as a function of time and this is known as the velocity potential function and just like with the stream function we had relationships for the velocity for potential we have u is this one's a little easier to remember because u goes with x partial phi partial x v is equal to partial phi partial y and also there's no sign change here whereas there was for the stream function so that makes it a little easier to remember and what we can say is that lines of constant phi or potential are potential lines so what we're going to do in the next segment we're going to explore this little bit further we're going to go back to the velocity field that we looked at when we derived a stream function we're going to derive the velocity potential for that same velocity field so that's what we'll do in the next segment