Anderson Video - Continuity Equation- Moving Fluids and Traffic

Professor Anderson
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>> Hello, class. Professor Anderson here. Another learning glass lecture on physics. Let's talk about moving fluids and see how this is very much like traffic. Okay. The cars on a freeway very much act like a moving fluid. Okay. Let's draw a picture of a pipe. Here's a pipe. And if we think about some slab of water moving through this pipe, then we can say that slab of water is going to have a mass m1, it's going to have a speed V1. But it's got to come out the other side of the pipe and so mass, m2, will have speed V2. We can also say that the pipe is cross sectional area A1 at this end, A2 at that end. And it occupies some length. That slab of water occupies delta x1 down here and delta x2 up there. What can we say? Well, let's say there's no holes in the pipe anywhere so water doesn't leak out. And there's not extra water coming into the pipe. And so the mass of water leaving has to be in fact equal to the mass of water entering. Those are equal. Okay. But we know what the mass of water entering is. It's m1. The mass of water leaving is m2. m1 is a density times the volume so this becomes density of the water times the volume of that slab which is A1 delta x1. The mass of m2 is going to be the density of the water times A2 delta x2, right. We just did density times volume. But if the water is moving at speed V1, then I know that delta x1 is just V1 times sum delta t. And delta x2 is going to be V2 times sum delta t. And now look what happens. The density of the water doesn't change. Ideal fluid, we said, is incompressible, meaning the density doesn't change. And this is very good assumption for water. Water has basically the same density no matter if you're pushing on it or not. And so rho drops out. Delta t is going to be the same, pick some amount of time on your stopwatch. And we get a very nice relationship between these two things. A1 V1 equals A2 V2. And this is known as the equation of continuity for fluids. And it really means nothing more than the mass is conserved. Water entering has to equal water leaving. If I constrict one of these areas, then the velocity will change according to this equation. And so now you can probably see the analogy between fluid flow and traffic. Let's draw a four-lane highway. Which we don't have many of these in Southern California anymore. Right. They're all like twelve, eighteen, some huge number of lanes. But let's say in the old days we had a four-lane highway. Okay. And you're driving along in your car. And you're moving along at speed V1. This is your cross sectional area. It's really a length but it's okay, right. What's the cross sectional area of the pipe is now the freeway. And let's say all of a sudden there's a crash and they rope off this section. So this is an accident in here and they've got fire trucks, and ambulances, and all sorts of things, and they rope it off. And so all of a sudden the four-lane freeway becomes a one-lane road. So you know what happens. All the cars stack up here and then they have to start merging over, okay. And there's lots of cars and lots of density. But as soon as they get into this region, they start moving at a speed, V2. And V2 is now given by this equation right here. What is it? It's just V1 times A1 over A2. So if this was four-lanes, then A1 would be 4. If this was one lane, then A2 would be 1. Your speed when you get into that one-lane region would in fact be four times faster. And you've probably all experienced this, right. You move very slow until you get next to the crash site and all of a sudden you start moving very quickly and the cars space out quite a bit. Okay. Same idea. The equation of continuity says number of cars that are going to pass here per second has to equal the number of cars that are going to pass here per second. And so if there's only one rho, they have to be going four times as fast. Questions about that stuff? When you have a hose and you are watering your garden, the hose has some diameter to it and you're going to have water shoot out of the hose. But let's say you want the water to shoot out at a much faster speed out of your hose. What do you do? What was your name again? Martin? >> Yeah. Martin. >> Martin, what do you do? >> If I want it to shoot faster? >> Yeah. >> Tighten one end. >> Yeah. You put your thumb over the end, right. So effectively what you've done is you have cut off some amount of the hose by putting a block right there. This is your thumb. And so now water only comes out of this region and it comes out a lot faster. And it's the same idea. You're constricting the area which means that it has to come out faster.
>> Hello, class. Professor Anderson here. Another learning glass lecture on physics. Let's talk about moving fluids and see how this is very much like traffic. Okay. The cars on a freeway very much act like a moving fluid. Okay. Let's draw a picture of a pipe. Here's a pipe. And if we think about some slab of water moving through this pipe, then we can say that slab of water is going to have a mass m1, it's going to have a speed V1. But it's got to come out the other side of the pipe and so mass, m2, will have speed V2. We can also say that the pipe is cross sectional area A1 at this end, A2 at that end. And it occupies some length. That slab of water occupies delta x1 down here and delta x2 up there. What can we say? Well, let's say there's no holes in the pipe anywhere so water doesn't leak out. And there's not extra water coming into the pipe. And so the mass of water leaving has to be in fact equal to the mass of water entering. Those are equal. Okay. But we know what the mass of water entering is. It's m1. The mass of water leaving is m2. m1 is a density times the volume so this becomes density of the water times the volume of that slab which is A1 delta x1. The mass of m2 is going to be the density of the water times A2 delta x2, right. We just did density times volume. But if the water is moving at speed V1, then I know that delta x1 is just V1 times sum delta t. And delta x2 is going to be V2 times sum delta t. And now look what happens. The density of the water doesn't change. Ideal fluid, we said, is incompressible, meaning the density doesn't change. And this is very good assumption for water. Water has basically the same density no matter if you're pushing on it or not. And so rho drops out. Delta t is going to be the same, pick some amount of time on your stopwatch. And we get a very nice relationship between these two things. A1 V1 equals A2 V2. And this is known as the equation of continuity for fluids. And it really means nothing more than the mass is conserved. Water entering has to equal water leaving. If I constrict one of these areas, then the velocity will change according to this equation. And so now you can probably see the analogy between fluid flow and traffic. Let's draw a four-lane highway. Which we don't have many of these in Southern California anymore. Right. They're all like twelve, eighteen, some huge number of lanes. But let's say in the old days we had a four-lane highway. Okay. And you're driving along in your car. And you're moving along at speed V1. This is your cross sectional area. It's really a length but it's okay, right. What's the cross sectional area of the pipe is now the freeway. And let's say all of a sudden there's a crash and they rope off this section. So this is an accident in here and they've got fire trucks, and ambulances, and all sorts of things, and they rope it off. And so all of a sudden the four-lane freeway becomes a one-lane road. So you know what happens. All the cars stack up here and then they have to start merging over, okay. And there's lots of cars and lots of density. But as soon as they get into this region, they start moving at a speed, V2. And V2 is now given by this equation right here. What is it? It's just V1 times A1 over A2. So if this was four-lanes, then A1 would be 4. If this was one lane, then A2 would be 1. Your speed when you get into that one-lane region would in fact be four times faster. And you've probably all experienced this, right. You move very slow until you get next to the crash site and all of a sudden you start moving very quickly and the cars space out quite a bit. Okay. Same idea. The equation of continuity says number of cars that are going to pass here per second has to equal the number of cars that are going to pass here per second. And so if there's only one rho, they have to be going four times as fast. Questions about that stuff? When you have a hose and you are watering your garden, the hose has some diameter to it and you're going to have water shoot out of the hose. But let's say you want the water to shoot out at a much faster speed out of your hose. What do you do? What was your name again? Martin? >> Yeah. Martin. >> Martin, what do you do? >> If I want it to shoot faster? >> Yeah. >> Tighten one end. >> Yeah. You put your thumb over the end, right. So effectively what you've done is you have cut off some amount of the hose by putting a block right there. This is your thumb. And so now water only comes out of this region and it comes out a lot faster. And it's the same idea. You're constricting the area which means that it has to come out faster.