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Volume flow rate and equation of continuity
Everything we've done so far has been stationary fluids, or static fluids, and we've been dealing with static pressure. We were trying to figure out what happens when everything's in a steady state. Now let's work on what happens when the fluid is actually moving. Let's imagine a pipe. Let's say one end of the pipe has a larger area than the other end, or at least a different area. So this is one end of the pipe, and this is the other end of a pipe. It's filled with some fluid, some liquid, actually, in our example, so there's just a bunch of liquid in this fluid. Let's say this area at the entrance is called the area in. That's the area of the opening into the pipe, and let's call this area out. It's the area of the opening coming out of the pipe. Let's think about what happens if this liquid is actually moving. Let's say it's moving into the pipe with the velocity V in. Let's think about how much volume moves into the pipe after T seconds. After T seconds, if you think about it, you'd have this much area. If you think about what was right here, it will then be moved to the right by how much? We could just go back to our basic kinematic formula: distance is equal to rate times time. The distance something travels equals velocity times time, so after T seconds, whatever fluid was here, it would have an area of about that much. Whatever fluid was there would have traveled how much to the right? It would have traveled-- let's assume that the pipe doesn't change too much in diameter or in radius from here to here. It would have traveled velocity times time, so V in times time. It could be meters or whatever our length units are. After T seconds, essentially this much water has traveled into the pipe. You could imagine a cylinder of water here. Once again, I know I made it look like it's getting wider the whole time, but let's assume that its width doesn't change that much over the T seconds or whatever units of time we're looking at. What is the volume of this cylinder of water? The volume-in over the T seconds is equal to the area, or the left-hand side of the cylinder. Let me draw the cylinder in a more vibrant color so you can figure out the volume. So it equals this side, the left side of the cylinder, the input area times the length of the cylinder. That's the velocity of the fluid times the time that we're measuring, times the input velocity times time. That's the amount of volume that came in. If that volume came into the pipe-- once again, we learned several videos ago that the definition of a liquid is a fluid that's incompressible. It's not like no fluid could come out of the pipe and all of the fluid just gets squeezed. The same volume of fluid would have to come out of the pipe, so that must equal the volume out. Whatever comes into the pipe has to equal the volume coming out of the pipe. One assumption we're assuming in this fraction of time that we're dealing with is also that there's no friction in this liquid or in this fluid, that it actually is not turbulent and it's not viscous. A viscous fluid is really just something that has a lot of friction with itself and that it won't just naturally move without any resistance. When something is not viscous and has no resistance with itself and moves really without any turbulence, that's called laminar flow. That's just a good word to know about and it's the opposite of viscous flow. Different things have different viscosities, and we'll probably do more on that. Like syrup or peanut butter has a very, very high viscosity. Even glass actually is a fluid with a very, very high viscosity. I think there's some kinds of compounds and magnetic fields that you could create that have perfect laminar flow, but this is kind of a perfect situation. In these circumstances, the volume in, because the fluid can't be compressed, it's incompressible, has to equal the volume out. What's the volume out over that period of time? Similarly, we could draw this bigger cylinder-- that's the area out-- and after T seconds, how much water has come out? Whatever water was here at the beginning of our time period will have come out and we can imagine the cylinder here. What is the width of the cylinder? What's going to be the velocity that the liquid is coming out on the right-hand side? Capital V is volume, and lowercase v is for velocity, so it's going to be the output velocity-- that's a lowercase v-- times the same time. So what is the volume that has come out in our time T? It's just going to be this area times this width, so the output volume over that same period of time is equal to the output area of this pipe times the output velocity times time. Once again, I know I keep saying this, but this is kind of the big ah-hah moment, is in that amount of time, the volume in this cylinder has to equal the volume in this cylinder. Maybe it's not as wide, or something like that, but their volumes are the same. You can't get more water here all of a sudden than what's going in, and likewise, you can't put more water into the left side than what's coming out of the right side, because it's incompressible These two volumes equal each other, so we know the area of the opening onto to the left hand of the pipe times the input velocity times the duration of time we're talking about is equal to the output area times the output velocity times the duration of time we're talking about. It's the same time on both sides of this equation, so we could say that the input area times the input velocity is equal to the output area times the output velocity. This is actually called in fluid motion the equation of continuity, and it leads to some interesting things. We'll do some problems with it in a second. One thing that I want to introduce at this point as well is what is the volume per second? Because this is also something we're going to deal with in a second, probably in the next video, because I'm about to run out of time. We said that in T seconds we have this amount of volume coming in and it's the same coming in as coming out. So what is the volume per second? It's this big capital Vi per amount of time, and we call that flux. We'll learn a lot about flux, especially when we start doing vector calculus, but flux is just how much of something crosses a surface in an amount of time. It's how much a volume crosses a surface in an amount of time. So in this case, the surface is the left-hand side of cylinder. And we're saying how much crosses in amount of time? We figured out it's that input volume, which crosses in every T seconds, and this is called flux. You've probably heard of the flux capacitor in Back To The Future, and maybe we can think about what they were trying to hint at. Let's see if we can use flux and these ideas to come up with some other interesting equations. We know that the volume per time is equal to flux. This is a big V. V is equal to flux, and actually the variable people generally use for flux is R. Of course, it's in meters cubed per second. That's its unit. We also know that the input area times input velocity-- that's a lowercase v-- is equal to the output area times output velocity, and this is called the equation of continuity. It holds true whenever we have laminar flow. Actually, I'm about to run out of time. In the next video, I'm actually going to use some of this information to figure out how much power is there in a system where we have fluid going through a pipe. See you soon.