 Okay, we're now going to solve an example problem involving a control volume moving at a constant velocity. So, I'll begin by drawing out the problem itself. Okay, so what we have here is we have a water jet and the jet of water is flowing from the left to the right in this direction here and it is impinging upon a blade and that blade is on rollers and consequently it is free to move to the right due to the fact that this jet or water jet is impinging upon the blade and so what we are told to solve for in this program in this problem we're told to find the force so we're told to find the force on the vein in order to keep you constant so essentially what we're told is what is the force required on the vein so this is our vein here there will have to be some force and we don't know exactly which direction is going to be moving in but a force something like that let me clean it up a little bit that's not the nicest vector but we have some force that we're looking for and the purpose is to find it for you to be constant and constant consequently that would be a case of a control volume moving at constant velocity because you can probably figure out we're going to put the control volume around the vein things that are given we're told that the velocity of the cart to begin with is 10 meters per second and that's the velocity that we want to maintain and the jet is coming in at a higher velocity it's coming in at 30 meters per second so this is a case where we will be doing analysis using a control volume moving at constant velocity so let's take a look at the solution okay so what we have here this is what our problem looks like we have the cart is moving at velocity you and I'll draw that as a vector going in that direction the jet is coming in from the left now what we're going to do we need to specify the control volume and for this problem what we're going to want to do is have the fluid crossing the control volume perpendicular to the control volume itself so we will draw the control volume like that so that it is perpendicular to the fluid streams and then I'll come over here the control volume I'll bring it right down to the ground but it's not going to go to the ground exactly and then I will bring it up so this is our control volume that we will define and what I'm going to do given that we have fluid crossing this boundary and this boundary I'm going to call this control surface one and this will be control surface two and then the last thing well another thing that will do remember this is a control volume that is moving and so it's moving we have a moving reference frame will give that a little action a little why and what we're going to do just that you can picture what's going on we're going to draw a little person on here because that person is moving with the control volume and then finally we're going to look for a force in order to maintain the velocity you to constant value and we will have a reaction force in the y direction and a reaction force in the x and I've just drawn them in that way positive x positive y we'll have to see by the sign that we get when we're done which direction they actually go in which is kind of a typical thing always doing statics so we have the fluid coming in the basic equations that we have the basic equations that we'll be using will be a conservation of mass or continuity and the momentum equation and so those will be the two basic equations a few assumptions let me write out the assumptions before we jump into the equations okay so a few assumptions that we have we have steady flow so even though the control volume is moving it's not accelerating because the force is holding it so that it only moves at a constant velocity and the conditions coming out of the jet itself from here we're not told anything is changing there so as far as we're concerned we can assume that to be steady flow coming in uniform flow so what does that mean what that means is that we're going to assume that this is kind of a fixed velocity profile if you're to look at the velocity profile coming in it is uniform it it's not a velocity profile that would have some sort of parabolic or or different type of shape like that we're going to assume it to be a uniform velocity profile sometimes we call that the top hat profile because it would be flat neglect body forces next thing density is constant so incompressible flow is what we're dealing with and the final assumption is the v1 is equal to v2 and so what that is implying the magnitude of v1 is equal to v2 so that would be the velocity of the fluid coming in at control surface one and the velocity or the magnitude of the velocity of fluid leaving control surface two okay so those are the assumptions what we're now going to do let's move ahead and start working with our basic equations okay so we have linear momentum and we have continuity so those are the basic equations continuity basic equations we have let's start with continuity and applying the assumptions that we wrote out we can eliminate some of the terms so okay so to begin with we have steady flow so this guy is gone and what we're left with is an integral across the two control surfaces if we look back here we have control surface one and control surface two so we apply the continuity equation across those two control surfaces and let's expand that out and I'm going to write the velocity vector with XYZ so that's in our with respect to the control volume reference frame I'm going to put it with a little subscript one and then for control surface two we have that now writing out the velocity vector or the yeah that'll be the velocity vector at control surface one what we have looking back and remember we have to we have to deal with velocity with respect to the control volume so we have a situation where we have V coming in and the control volume itself is moving at U so the velocity coming across here with respect to the control volume is going to be the difference between those two so we will have V minus U and then we can express that on inlet both components are moving in the X direction and so we get this I can write it out in the I component then for the velocity leaving this is for one of the assumptions we had we said that the magnitude of the velocities were the same so looking back at our drawing we know the angle here is 60 degrees so we can use trigonometry to figure out what the velocity component would be leaving so that would be the X component and the Y component and there we've made use of the fact that the magnitude of V one is equal to magnitude of V two one of our assumptions so what we can do is we can put this together and what we get for continuity and remember we need to be careful with the sign here so for control surface one let me just sketch this out because this is sometimes an area where students have difficulty so for control surface one let me draw the area remember the area vector is always going to be going outward from your control surface so that's the control surface but we have a velocity that is moving in this direction actually it's the V minus U and and so with that this is going to be a negative and then for control surface two what we have we had a control surface that was angled again the area vector is moving in that direction and then we have the well I'll just draw it as a vector but our velocity vector is going to be going in the same direction and so that will be a positive so consequently what we get from the continuity equation is the following and from this density cancels out because we're dealing with incompressible flow and we don't know anything about area two so I'll just leave that as area two but we get the magnitude of XYZ at interface one control surface one times area one is equal to the magnitude of velocity at control surface two times area two so that's what continuity gives us we're going to park that and then move on to take a look at the linear momentum equation in the next segment