Fluid Flow & Continuity Equation - Video Tutorials & Practice Problems
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1
concept
Fluid Speed & Volume Flow Rate
Video duration:
6m
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Hey, guys. So in this video, we're going to start talking about flow continuity, which is one of the most important topics in fluid flu. Let's check it out. All right. So first I wanna go over the two main terms that deal with how quickly a fluid flows, and you need to know the difference between the two. It could be tricky because they sound similar. So you have fluid speed and volume flow rate, and it's kind of tricky because speed and rates sort of tend to mean the same thing. Now fluid speed is just basically distance Over time, it's meters per second. How many meters? How many meters will, let's say, a water molecule travel in the pipe over the course of one second, So it's just Delta X over Delta T or distance over time. So let's say we have two pipes and in this pipe water, a little water molecules goes from here to here in one second and another in the other pipe water molecule goes from here to here in one second. Which one has the greater fluid speed? Hopefully you got that V two would be greater than V one because this is a greater distance in the same amount of time. Okay, so just has to do with how quickly is water or whatever Fluids moving. So if you have this one moving like this and the other one moving like this, getting ahead of it, uh, than the bottom one is faster volume flow rate is a little bit different. It's not distance, it is volume hands the name volume, right. So you can see in the units meters per second and then this is meters cubic meters per second. This is not the distance but the amount of volume over time. Now, before we get confused, noticed that I wrote this as a lower case V which is speed and this is uppercase V, which is volume. Okay, don't get those mixed up. So the idea is, let's say you have two pipes and one is is thicker than the other. So in this pipe of molecule, water goes from here to here in one second and then in this pipe you're gonna get the same thing. Molecule water goes from here to here in one second as well, so the distances and the times are the same so V one equals V two. However, if you have a ton of water going through this pipe, even though they're moving at the same fluid speed, there's a lot more water that is traveling from here to here. There's a lot more volume of water here, So because this has the greater volume, we're going to say that this here has the greater volume flow rates, which is given by the letter. Q. So I'm gonna say that Q two is greater than Q one. That should make sense. It's a bigger pipe, so it carries Mawr volume of water per second if the speed is the same. Okay, I want to do one more thing with this equation here. I'm gonna rewrite Delta V over Delta T over here, and I want to remind you the volumes of three dimensional measure and in a pipe like this, volume would be the cross the cross sectional area here, times the distance. Okay, so let's say a little distance here. This volume here would be the cross sectional area a times this little distance. Let's call this distance Delta X. So the volume there is a Delta X, so I can rewrite Delta V as a cross sectional area, which is constant in this pipe times, Delta X, and notice that I end up with Delta X over Delta T, which I can rewrite as a V. This is just speed. Okay, so I'm gonna put this over here, and I need you to know that fluid speed is Delta X over Delta T and volume flow rates is Q equals Delta V over Delta T, but can also be written as a little V. Okay, this is little V. This is speed. Let's do a quick example to highlight the difference between these two. So long horizontal pipe has a two square meter, a cross sectional area. So let's draw a pipe. And that means that this area here is too. And it says it takes, um, it takes water five seconds to traverse the travel an 80 m segment of the pipes. Let's say that from here to here, water moves. This is a distance or delta X of 80 m, and it takes Time T or delta t of five seconds. And we wanna know what is the fluid speed. Fluid speed is just little v Delta X over Delta T. This is 80 divided by five, which is 16 meters per second. If you wanna know the volume flow rates, that's Q, which is volume over time now. I don't have the volume. I could calculate the volume, but I don't have to because I also know that I can rewrite this as area times speed every time speed I have. The area is too square meters and the the speed is m per second. Notice what happens here. Meters square with meters become cubic meters per second. You multiply the two numbers. You end up with 32 so 32 cubic meters travel every second. So that's how those two things work. And the reason why this distinction between the and Q is important is because it's gonna play into the next thing we're gonna talk about, which is flow. Continuity and flow continue. It is a pretty important principle, and it's the idea that because an ideal fluid is in compressible its volume flow rates volume flow rates, which takes letter, remember Q. It is never going to change, so I need you to remember that Q never changes. Okay, so for example, let's say that you are looking through a pipe like this that changes area, right, so you have area one and then you have area to, which is obviously smaller than area one. And let's say that there's a certain amount of volume that goes through this cross sectional area, right? So if you sort of pick up points in the pipe, a certain amount of volume goes to that point. Well, if you pick a point a second point at this thinner part of the pipe here, the same amount of volume has to go through that second point every second as well. So the rate has to be the same. But now there's a much smaller area. So how do you get the same amount of water per second to go through this piece? If this piece is much skinnier, the only way to do that is if the water ends up moving much faster. And that's the idea of continuity that the flow is continuous and it never changes. So if the area becomes smaller than the speed has to increase to make up for that, Okay, so let's right the south real quick and by the way you may have done this if you've ever played with the with a water faucet, right? So there's a little bit of there's some speed with which the water is coming out. But then if you cover this maybe, like with your thumb, I'm gonna draw an ugly thumb here. Um, if you drop cover this with your thumb, that looks terrible. Let's put a little hair there. Um, what's gonna happen is that the water is gonna now come out faster because you cut, you closed it. Okay, Um, so que never changes, Remember, Q can be written as the volume change in volume over time. It could also be written as area times speed. Okay, you can also be written as area time speed. So the idea, um, in this, by the way, this is called the continuity equation continuity equation and the fact that a V's constants means that if the area changes than the velocity or the speed has to change in the opposite direction, so greater area would mean slower speed. So if area changes speed changes, and this segment here that I circled can be sort of extracted out of the equation, written like this area. One speed one equals area too speed too. And that's because they ve never changes. So if one goes up, the other one goes down. The product of those two is Constance. Last point I wanna make before we make it do an example here is that pipes are usually going to be cylindrical. So the area like this area here, If this is cylindrical, he's gonna be pi r squared. It's the area of a circle. Let's go quick. Example. So I got a garden hose with radius to and at the end, there's a nozzle of radius one. So it's something like this radius to, And then at the end, there's a nozzle with radius one. So here, I'm gonna call this art. One is two centimeters and our two is one centimeter. Um, water flows into it with 2 m per second. So speed one is 2 m per second and we wanna know what is the speed over here? So what is speed two? So this is part eight. So how do we solve this? Well, whenever we have a pipe and the areas changing, we're always gonna be able to use this equation to figure out one of the speeds or one of the areas. Okay, so we're gonna go straight for this. We're not gonna use the sort of bigger equation has all the variables we're just gonna use, um, this one. So we're gonna write that a one V one equals a TV, too. I'm looking for V two, so I just have to move a to to the other side. So v two is gonna be V one times a one over a two. Okay, The one we have, it's 2 m per second. A one. Remember, this is a, um it's cylindrical pipe. The way I know that is because it says here it's got a radius. So the area is gonna be pi r one, um squared. Divided by pi r two squared. Notice that the part the pious cancel. That's cool. And then I have 2 m per second. The radio ir, um, two and one. So two centimeters in one centimeters. Uh, now you might be thinking shouldn't I Can't Shouldn't. I shouldn't replace or convert from centimeters. 2 m. You don't have to because they're just gonna cancel each other out. So for this part. You don't have to convert anything. Okay, so this is gonna be two square, which is 44 times to eight. So 8 m per second is the speed with which water will come out here. So let me write that here. 8 m per second. And then for parts B. We want to know in how many minutes does does it fill up a 350 leader bathtub? So this is giving us the volume 3 50 leaders, and it's asking for the time, but it's asking for the time in minutes. So it's gonna happen, is you're gonna find a time in seconds and you're gonna convert it to minutes at the very end. Let me get out of the way, and we're gonna keep going here. So what do we do? Well, now we're gonna go back to this big equation, and this big equation sort of has all the variables, and you can see how it has a delta t in it. Right? So I'm gonna write the whole thing just real quick. Delta V over Delta T I like to think of this equation. I liked to sort of remember this whole thing together. Okay, notice that what I have here is the definition of Q the two definitions of Q and the fact that a one view one equals a TV to all into one equation. Now, there are three equal signs here. There are four parts to this. So what we do is we just pick two of them. I want times. I'm gonna pick this in this, and I'm gonna say the Delta V over Delta T equals a one view one I could have said equals a to V to just the same. All right, so let's start plugging some stuff so the volume is going to be the volume is going to be 3 50 leaders divided by time, which is what I want. Area is gonna be pi r one square. Now, I actually have to convert because I'm gonna be calculating stuff. There's nothing to cancel centimeters with, so I have to convert this into meters and the R one r one is two centimeters or 20.2 m and that's gonna be times the first velocity, which is 2 m per second. Okay. And if you put this whole thing in a calculator. If you put this whole thing in the calculator, you get that. This is 0.25 um, cubic meters per second Notice that I have meters squared and then meter. So that's three m's. So that's cubic meters. Okay, um, that's what that's gonna be, but I'm looking for time. So I'm gonna move time up here, and then this guy is going to come to the bottom of the 3 50. Okay, so please do this very carefully. It's gonna come out like this. Delta T equals 3 leaders divided by 00025 meter cube, divided by seconds, getting tightened there. But we'll make it work. Now Notice that you have leaders and which is a volume and meter cube, which is also volume. But they can't directly cancel because 1 m is not equal to one. I'm sorry. One leader does not equal one cubic meter. So hopefully you remember that one cubic meter Put this a little closer. One cubic meter is actually 1000 leaders. Okay, So what I can do is I can rights over here. That one cubic meter is leaders and in doing this. What happens? I can cancel this and this. Okay? And if I multiply this whole thing here, I get that This is 140 seconds. Okay? But we want this in minutes. Last step is to divide by 60 seconds, because that's what one minute is. And long story short, We end up with 2.33 minutes. So this water pipe would take 2.3 minutes to fill up this bathtub. And that's it for this one. Let's keep going.
2
concept
Flow Continuity
Video duration:
11m
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Hey, everyone. So now that we've talked about the difference between fluid speed and volume flow rates. In this video, we're going to discuss an important consequence of that, which is called fluid or flow continuity. This happens whenever you have some liquid that's traveling in a region in which the area is going to change like a pipe that gets skinnier or fatter or something like that. All right, we're gonna discuss a really important equation and then I'll show you an example. All right. So continuity basically says that because ideal fluids are incompressible because their densities can't, they can't get more or less dense and they can't get squeezed. Then the volume flow rates of a liquid has to remain the same. So volume flow rates, which remember is given by the letter Q is never going to change. So we can write this as an equation and we can say that remember Q which, which is given as delta V over delta T. We saw that in the last, in the last couple of videos and we could also rewrite this in terms of area times speed, all these equations equal to each other that has to remain constant. All right. So this important relationship here that I've highlighted is oftentimes called the continuity equation. So let me go ahead and show you in this diagram, what's going on here? If you have some kind of a a pipe or you have some water that's flowing through this pipe over here and it's got a cross sectional area, a one. And let's say the fluid speed is traveling over here. Then basically what we know is that area times speed is going to give you a volume. And I'll highlight this in yellow over here. So you have uh some amount of volume that's flowing through this pipe. Let's just go ahead and call this 10 L uh just to give it some numbers, right. What continuity says is that if you have 10 L of water that's flowing through this pipe every second, then later on, if the pipe changes its area, it's cross sectional area, you still have to have 10 L that is traveling through the pipe, you can't have 5 L because then where all the extra water go and you also can't have 20 L because then all of a sudden you just created a bunch of liquid out of nowhere. So basically all of this stuff that's flowing through this pipe has to remain flowing at the exact same rates. The volume flow rate always has to remain the same. Ok. Now, what we can see here is that in this pipe, the cross sectional area has gone down this cross sectional area over here. A two is much less. So what this equation tells us this continuity equation is if the area changes like the area goes down, then the only way that you can have this cue that remains the same is if the speed increases. So in other words, the speed in this skinnier section of the pipe has to go up like this. All right. So, and also this works vice versa. If the area goes up, the speed has to go down. So basically what continuity says is that if the area changes, then the speed also has to change. All right. And here's probably the most important equation that you need to know the area one times V one is gonna equal area two times V two. OK. This is basically the equation that you're going to use when you talk about continuity. All right. The last thing I want to mention here is that pipes are usually usually cylindrical, which means that they cross sectional areas are going to be a or a pi R squared. All right. Now, if you've ever sort of messed with like a garden hose or something like this, you've probably sort of like stuck your thumb at the end of the garden hose, right? So you've got some liquid that's coming out, you see your thumb over here. Uh This is gonna be your thumb like this. Um and basically what happens is the water comes shooting out much, much faster when you do that. And that's basically what's going on here, right? So you have some cross sectional area, let me go ahead and do this in, in uh in red. So you've got a and then what happens is when you plug your thumb over the hole, you've basically made the opening in which the water can come out much smaller. And so the velocity increases as a result of that, right? So it kind of, it's kind of like a real world example. All right. So let's go ahead and jump into our example. Here, we have a garden hose with a radius of two centimeters. So in other words, this radius over here, I'm gonna say that this R one here is two centimeters. Um And then what happens is it's flowing, you have water that's flowing at 2 m per second. So in other words, we have that V one is equal to 2 m per second. Then what happens is the nozzle sort of gets a little skinnier and we have a radius of one centimeter. So this is gonna be R two equals one centimeter. And we want to do is we want to figure out what the speed of the water is in the nozzle. So basically, once that pipe gets sort of once the guard hose gets skinnier into the nozzle, uh what is the speed at which the liquid comes out that's going to be V two over here. All right. So it's pretty straightforward, right? So we've got V one, we've got V two and we're comparing two different points within the same pipe. We know that the volume flow rate has to remain the same. And we can just go ahead and start off with our continuity equation. We can say that a one, V one is equal to a two V two. We're looking for V two. So basically what this means is that V two is equal to V one times area one over area two. Now remember this is a cylindrical pipe. So we've got something like this, right? So if you have some sort of like cross sectional area like this, the volume is gonna be this, this is basically all cylinders. Uh And so what this means here is that we can say that V two is equal to V one. And remember the cross sectional area is gonna be pi R one squared and then over uh area two is gonna be pi R two squared over here. Now, what happens is the pi is just cancel because they're gonna get divided and this is just gonna be uh I got V one which is 2 7 2 m per second. So it's 2 m per second. And then we've got uh R one square. Now, R one is equal to two centimeters over here. So we've got two centimeters squared, um, over one centimeter and all that's gonna be squared. Now, you don't have to convert this because remember whenever you have two units that are divided by each other, if they're not si units, but they're the same, it's ok because if you were to convert them to si, then it's just gonna end up being the same thing anyways. All right. So what happens is you can actually just keep this as centimeters over centimeters. It makes no difference whether you convert it or not. Now, what you're gonna get when you do this is you're gonna get that. This is equal to 8 m per second. All right. So you get 8 m per second and that is the answer to the first part. All right. So what happens here is you've uh shrunk the radius um by a half, but the speed actually went from two and it became 8 m per second and basically has to do with the area being proportional to R squared. Remember area it's proportional to R squared. So if the area doubles, um or if the radius doubles the area actually gets multiplied by four, right? So that's kind of what that has to do with, let's say, look at the second part here and the second part, we want to calculate, how many minutes does it fill up a 350 L bathtub? So, what does that mean? So remember this is just a volume here. So this volume is equal to 350. You're gonna use this garden hose here to fill up a bathtub. So you've got a volume of 350 L. And we're asked for how long and how many minutes? That's a time. So this is delta T and this is gonna be delta T in minutes. All right. So how do we do this? Well, if we look through our equation here, remember that Q is equal to delta V over delta T from the volume flow rate is volume over time. So that's the equation that sort of has all of this stuff sort of wrapped up together, right? So we have the Q is equal to delta V over delta T. Now, what I like to do here is remember that this delta V over delta T is also equal to area times volume. But what I like to do is I always sort of just like to continue on this equation and just write the continuity equation, this equals area one V one. But it's also the same thing as area two V two. Remember those things are equal to each other, right? So if Q is the same, then a one V one has to equal A two V two, all these things are equal to each other. Now there's lots of equal signs here. It turns out you don't need all of them because really you can pick any one of these points as long as you know, the area and the, and the speed and we actually know the areas and speeds for all of these points over here. So basically what you can do is you can go ahead and just sort of pick out these two equations or these two terms or you can also use this. So on one right here that you could also use this and you would get the exact same answer later on. All right. So I'm just gonna go ahead and pick a one V one and I'm just gonna write this down over here. So delta V over delta T is equal to a one V one. OK. So what's my delta V? It's just the change in the volume. I'm gonna fill up a 350 L bathtub. So that's gonna be my volume here. So I've got that 350 L divided by delta T. This is my target variable here. This is gonna equal a one times V one. The A one is just equal to pi remember this? A one here is gonna be pi times R one R one squared. Uh And this is gonna be times the one I actually went ahead and start plugging in some numbers, right? So I've got pi times this is gonna be a 0.02. Now, I have to plug it in in terms of si because I'm not gonna be able to cancel it with something else that's in centimeters. I only got one r uh variable over here. So I have to plug this in in meters. So it's gonna be 0.02 m squared times and this is gonna be 2 m per second. All right. Notice how, when you multiply these things, what you're gonna get when you work this out is you're gonna get something that's in terms of meters cubed per second. This ends up being 0.00 25 m cubed per second over here. All right. And you're gonna get 350 L 350 L divided by delta T. All right, now, I'm looking for delta T. So basically I have to swap these two and switch places. So I'm gonna rewrite this and I'm gonna say that delta T is equal to, this is gonna be 350 L divided by 0.0025. This is meters cubed per second. All right. Notice how I end up with volume on the top and I also end up with volume per second on the bottom. But these things are not the same. We have leaders and then meters cubes. So hopefully guys realize that I'm gonna have to convert one of them in order to be able to sort of cancel out and then divide right. I can't go ahead and divide now because 1 L is not necessarily 1 m cube, 1 L is not 1 m cubes, right? So what I have to do is because these two things are different. I have to insert a conversion factor. The conversion factor. Remember is that if I wanna cancel out liters and I wanna get end up with, I wanna end up with liters in the bottom. And if I wanna convert this to meters cubed like this, then 1 m cubed is equal to 1000 L. So basically what you end up with is that you end up with um equals 0.3 50 m cube divided by 0.0025. And that's in meters cubed per second. Now, now you can notice here that we actually can divide and when you divide this, the meters cubed is gonna cancel. And anyway, what you end up with with for delta T is 100 and 40 seconds which when you write this in terms of minutes, uh this is your final answer is gonna equal 2.33 minutes. All right. So that's how long it would take you to fill up this bathtub over here. All right, folks. That's it for this one. Let me know if you have any questions.
3
example
Continuity / Proportional Reasoning
Video duration:
4m
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Hey, guys. So in this example, I want to quickly show how to deal with proportional reasoning or proportional change questions involving continuity. Let's check it out. All right, so here we have water flowing A horizontal horizontal, cylindrical pipe, something like this. And it says water has a speed of V at point A. So let's say that this is point A and at this point, speed at a will be big V in point B has doubled the diameter. So somewhere over here, this thing grows to have double the diameter, right? So point B is somewhere over here be and the diameter. Let's say that the diameter of a will be, um d and a diameter of B will be twice that two d. And we want to know what is the volume? What is the velocity? The speed of water at this point, okay. And first, I want you to think about this, um, in conceptual terms, do you think the water will be faster or slower here and hopefully you pick that the the water will be slower. Remember if water is going into a tighter pipe part or a tighter segment of the pipe it's gonna go faster. Eso, if it's going to a wider section, is gonna go slower. And that's because water or fluid flow rates. Q. Which equals a times speed, is a constant. So if if the area increases, which it does here, the speed has to decrease so that the product a V, stays the same. Okay, so one way to think about this is if this is a to and this is a right and this grows to a four, this is 20. This has to decrease to a five so that this is still cool, so it should be slower, which means that it's not going to be the same. It's not going to be faster, so it's now down to whether it's one it Zvi over four or V over to, and what you can do is you can just write. You can write a one view one equals a to the to right, and we're solving for or I guess I could say a and A B right And we're writing. We're solving for VB. So VB is the first area, um, times, the first speed divided by the second area. Now the area of a cylindrical pipe is pi r squared so I can write pi r squared, divided by pi r squared times the first velocity which is the okay. Now I don't have the radio. I have the I have the, um the diameter, but diameter is half the radius. And if the diameter is doubling, that means that the radius doubles as well. So I can simplify this whole thing by saying I'm just gonna call this are and this is going to be too are Okay, So if the diameter doubles the radius doubles and all these questions whenever you have diameter pretty much in all of physics, whatever you see diameter, you're supposed to change that into radius. Okay, so one is double the other, so the pies will cancel, and I can say that a is our and then this guy here is to our times v. So look what happens. I have, um I have r squared this to hear becomes a 44 r squared. So the R squares canceled and you're left with V over four. Okay, so if the radius becomes twice as big, then the speed will become four times smaller And that's because the area depends on the square of the radius. So if the radius becomes twice as big, then the area becomes four times greater, which means that the speed has to go down by a factor of four. So the answer will be vey over. Four. Cool. These are pretty popular. Hopefully, this makes sense. Let's keep going.
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