 in this segment what we're going to do is we're going to take a look at one of the four general motion types of fluid and that will be fluid rotation and fluid mechanics is described by a vector omega so it's defined as being the average angular velocity of any two mutually perpendicular line elements and the way that we define it I mentioned it's a omega a vector and we will have a fluid rotation in the I direction one in the J direction and one in the K direction and the convention that we will use for the sign of it we will use the right hand rule which implies that counterclockwise is positive for the rotation so when we're looking at fluid rotation let's draw it a little schematic here so that might be our original fluid element now if it's going through pure rotation what we can do is that's what our object might look like through pure rotation so we're going to zoom in on that and we have our two orthogonal lines that are rotating we're going to zoom in on that and we're going to write out some terms and that will enable us to come up with a derivation for omega a little omega okay so here we have our deformed fluid particle on the right hand side and what we have are two different angles we have a delta alpha and a delta beta and those will quantify the rate of rotation and you'll notice I have a delta x and a delta y what we're going to be looking at is we're going to be looking at the rotation rate of omega o a that line and we'll also look at omega o b so we're going to take these one at a time that's not a very good omega let me erase that we're going to take these one at a time and then we're going to average those together because that's what we said it's the average angular velocity of any two mutually perpendicular lines so now let's go through and look at this and we'll come up with the equations that describe the rotational rate of a fluid particle so to begin with what we'll say is the v velocity at zero let's assume that we could have this rotating fluid particle translating so we'll give it a translational velocity v not so the origin could be moving at v not and the velocity at a looking back at the image here so we're looking at this point here a and we're looking at the v component in the y direction that would then be equal to whatever that translational velocity is plus and I'm going to use a Taylor series expansion here dv by dx multiplied by delta x so essentially what we're doing is we're looking at how is the v component of velocity changing as it goes from the origin to this point here a and that's how we can express that we're now going to use that and we're going to try to figure out the rotational rate of line zero a so origin is there a is there we want to figure out that rotational rate so what we can do we can write this out as a limit and it is going to be the change in alpha divided by the change in T so if we look back alpha is just this angle so we're looking at the change of that angle with respect to time which is the angular rotation rate so that's pretty simple kind of makes sense but what I'm now going to do is I'm going to substitute for delta alpha and I'm going to use the small angle approximation and doing this to keep the limit and using the small angle approximation I can say the delta alpha is delta eta divided by delta x and then I retain the delta T there and the other thing that I know is that delta eta can be expressed as being the velocity at that location multiplied by delta x what I've done here is I've taken my Taylor series expansion that is the differential or the increase in velocity at a with respect to the origin and then I multiply that by delta t and that's how far a is going to move in a period of delta t so I can take this and I can substitute it in to this expression up here and when I do that I can re-express my angular rotation and so I have all those and when you look at this magically a bunch of stuff starts to cancel which is great and what we end up with is just dv by dx so that is the angular rotation for line 0a now I'm going to go through the same sort of logic looking back at the diagram I'm going to look at a line 0 to b so let's go through the same process and in this case I'm going to be looking at the u component of velocity sorry that should be u at the origin is equal to 0 and then u at b is that origin velocity whatever the translational velocity might be multiplied by and this is basically a Taylor series expansion here now what I'm interested in is rotational rate of 0b so the line to the origin again I'm going to do the limit and that is just delta beta divided by delta t and I'm going to make the small angle approximation so delta beta can be expressed as delta zeta divided by delta y and then that's all divided by delta t now delta zeta I'm going to get that by knowing the velocity at b and I can get that from this term here and notice here I'm going to introduce a minus sign because it's going to the left and when I substitute this into my angular rotation expression I then end up with the following again just like before a bunch of the terms are going to cancel out oops I just disappeared my delta t that's gone that's gone that's gone that's gone what do we end up with we end up with a minus partial u partial y so that becomes the rotational rate of 0b from the origin to point b now what we're going to do remember we said that rotation was the average of the angular rotation of two perpendicular lines therefore omega z is going to be the average of those two rotational rates and I get that for the rotational rate in the z direction and I can go through the same sort of argument and rationalization for the other components of rotational rate and we get this so we get those and let's put them all together into our vector and so we get that describing the rotational rate of a fluid particle and we can also take advantage of a vector operator the curl which is in our review of vectors and so you can look at that if you want more information but when we look at the expression that we have for rotational rate what we find and if we compare it to the curl of the velocity vector omega is then just one half del cross v so that is the expression for the rotational rate of a fluid particle it is one of the four general types of translation or of motion that a fluid can be going through if you want to go back and see the other types I'm going to put a return box here so all you need to do is click on that and that will take you back and you can look at others or you can move on to other segments