 We're now going to continue on looking at different properties that we can evaluate with the velocity field and these are kinematic properties. The next one that we're going to look at is the volume rate of flow and this is something that we will use quite often in fluid mechanics and so we often denote volumetric flow rate or volume flow Q and then sometimes we'll put an over dot to denote rates. Now what we'll begin with is a surface or a body and we'll begin with coordinates and we have some arbitrary body here and let's imagine that we have a surface on this body and this surface what we're going to do we're going to draw a unit normal vector so that's n with a hat to denote the fact that that is a unit vector and if you recall if it is a unit vector and we take the magnitude of a unit vector that will be equal to 1 so there we have a unit vector it is normal to the surface and let's then imagine that we have some velocity vector v that is leaving the surface but it is not in the same direction as the unit normal and what we will do is we will specify the angle between the unit vector and v as being theta and this particular area that we're looking at we'll call that dA and this is over a larger surface S so this is basically a body with fluid leaving or entering so it's some arbitrary body we could have fluid coming in or leaving now when we're doing or computing the volumetric flow rate what we're most often interested in is going to be the component of velocity in the direction of the unit vector which is normal to the surface dA so we're usually interested we're interested in the component of v in the direction of n the reason is that velocity going parallel to the surface doesn't leave the body and consequently we're not interested in that we want to know the volumetric flow or the velocity of the fluid in the direction of that unit vector which is normal to the surface dA so the way that we evaluate that we use a mathematical operator for that that you've probably seen in your math courses and that is the dot product so we take the velocity vector v dot it into the unit vector and the definition of the dot product it's the magnitude of v multiplied by the magnitude of the unit vector which is one multiplied by cosine theta the cost of the angle between the velocity and the unit vector and what we get with this then is v cosine theta so that gives us the velocity component leaving the surface now what we want to do is we want to extend that to volume because we're interested in volume out or in so we will look at delta v out or delta v in and so what we're going to do we're going to take a look at another area so we'll call that dA and let's assume that there's some unit vector n and if we have some velocity here what we are most interested in is going to be the component of the velocity in the direction of that unit vector so what we're most interested in then is going to be the magnitude of this side here which from the equation that we looked at just a moment ago is going to be v cos theta assuming that theta is the angle between the velocity vector and the unit vector and if we're looking at this for a period of time to get volume flow it is then going to be multiplied by dt so what we can say here is that only the normal component flows in or out so only the normal component of the velocity vector is going to flow in or out of our surface that we're looking at and that's the one that we're most interested in so with that what we can write is delta v is v cos theta dt multiplied by the area dA and that would then give us the volume that is leaving or entering and that can be re-expressed in terms of the dot product that we looked at earlier now that's a delta v now what we want do we want to look at the elemental flow rate which that would be equal to delta q and delta q is going to be the elemental volume divided by time delta t so with that our expression can be rewritten in the following manner and notice the dt disappears in the limit of infinitesimal values so what we end up with for the volumetric flow rate into or out of a surface we would apply a surface integral over that entire kidney shaped object that we had so this entire surface we would apply a surface integral and the equation that we would be integrating in order to get the volumetric flow rate is going to be the velocity vector on the surface multiplied by a unit vector normal to the area on the surface multiplied by the area itself dA so you would integrate over the entire surface and from that you would then be able to determine the volumetric flow rate either going into or leaving a surface and so we use the dot product in order to enable us to do that now there is one thing that I should say here and the thing that I should say is that in our studies we will use the convention of n pointing outwards and will always be pointing outwards and so what does that mean what that means is flow out of a surface is positive and flow in is negative and that results due to the dot product that we had we had the v dot n and as a result of that if n is always pointing outwards from a surface and the velocity is moving in a similar direction that would be a positive flow rate out if the velocity vector is moving in the other direction that would give us a negative value because there we would have volume flowing into the surface but that is just the convention that we will use in fluid mechanics within this course.