 Okay, we're now going to work an example problem that involves the continuity equation. So this is mass conservation applied to a control volume. So what we're given, let me draw out a schematic of the problem to begin with. So what we have is an object. It's essentially a mixing chamber where we have a number of different streams either coming in or coming out. And we're told that this angle here for this stream is 60 degrees. And we're given the following information. So we're given that information and we are told to find v3, which is the velocity at area 3. So that is the unknown for the problem. And the control volume is specified here. The control volume is basically right around the box or the mixing chamber that we're looking at. So that is the control volume for the problem that we're dealing with. So this is a pretty simple case of the continuity equation. Looking at the vectors that we have for the velocity field coming in, v1 applies to area 1. And it is a positive 4 in the i direction. So that tells us that it is going in this direction. That would be v1. I'll draw it as a scalar with direction. And then v2, that's coming in across area 2. It's a negative, so it's going down. So what we know is that that is v2. So we have two streams coming in and we then know that the third stream is going to be leaving. It's going to be leaving at an angle because they show us that at 60 degrees. So what we're going to do, we're going to go through applying the conservation of mass equation. Another thing to note here is the density has been specified. So that means that we have a case of incompressible flow and it is steady. So if it's incompressible, the time rate of change term drops out of the continuity equation anyways. But that is the problem. Let's work through it and work on finding v3. So given that it's incompressible, steady flow, we can write the mass conservation equation as being this integral across the control surfaces. These are the three control surfaces where we have mass crossing the boundary. So I'm going to expand that out and I'll call it control surface one, two, and three for the three inlets or the two inlets in the one exit, I should say. Okay, so one thing that we can do right off the bat is looking at this. We have uniform flow. So if you recall earlier, we said that uniform flow you could approximate as rho vA taking the absolute value of that. VA is constant, so that's fine, velocity is constant and we know the areas. So we'll make that substitution for each of these terms here. However before we do that, let's take a look at the direction because remember we have the dot product in each of these and the dot product will let us determine whether it should be plus or minus, whether or not mass is coming into or leaving our control volume. We pretty much know in this problem because it's pretty explicit, but let's look at it explicitly just to be certain. So first of all for control surface one, the unit vector for the area dA1 would be in that direction and we were told that the velocity was going in that direction. So this would be a case where we're going to get a negative through the dot product. For control surface two, this was on the top. Again the area vector is going to be out in that direction. The velocity we were told was coming down, so again that's going to be a negative. And finally for area three, we have the area like that and although the channel is showing it coming out at an angle, what we will do, the area is this direction. So that is the vector of the area. The velocity itself however is going to be at an angle and those are both in the same direction and so this is going to come out as being a positive, but we have to take into account the fact that the velocity is going to be at an angle and we'll do that later on in the problem when we decompose things. So with that what we can do is we can rewrite the continuity equation in this manner and here the density is going to drop out of all of these equations. So what we'll do in the next step is we just plug in values and we'll solve four. What are we looking for? The only unknown here is v3 and we're going to get a scalar value that we then have to put into a vector component. So when we solve for this we then get the magnitude of v3 as 4.666 meters per second. Now what we can do, we do know that that is going to be at an angle of 30 degrees to the horizontal and so we can go through with trig relations and break that into a vector component having our vx and then a vy and what we then find for v3 expressed as a vector and that gives us the velocity and that was what we were after for this problem. So that's a pretty straightforward application of the conservation of mass or continuity equation using the control volume. Probably the trickiest part, it really isn't tricky, but you just have to be careful with the dot product in the way that you apply it and ensure that your areas are always pointed out and then whichever way the velocity vector is going you can then work through it from there. So that concludes the problem.