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19. Fluid Mechanics

Fluid Flow & Continuity Equation


Flow & Continuity Equation

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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.

Continuity / Proportional Reasoning

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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.