19. Fluid Mechanics
Fluid Flow & Continuity Equation
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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. Alright. So here we have water flow in 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. And 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. B and the diameter, let's say that the diameter of a will be, d and the diameter of b will be twice that, 2 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, in conceptual terms. Do you think the water will be faster or slower here? And hopefully you pick that the air, that the water will be slower. Remember, if water is going into a tighter pipe part or a tighter segment of the pipe, it's going to go faster. So if it's going to a wider section, it's going to 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 AV stays the same. Okay. So one way to think about this is if this is a 2 and this is a 10, right? And this grows to a 4, this is 20. This has to decrease to a 5 so that this is still 20. Cool. So it should be slower, which means it can't it's not going to be the same. It's not going to be faster. So it's now down to whether it's 1, it's it's v over 4 or v over 2. And what you can do is you can just write, you can write a one v one equals a two v 2. Right? And we're solving for, or I guess I could say a a and a b. Right? And we're writing, we're solving for v b. So v b is the first area, time, 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 v. Okay? Now I don't have the radii. I have the I have the, 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 r, and this is going to be 2 r. K? So if the diameter doubles, the radius doubles. And in all these questions, whenever you have diameter, pretty much in all of physics, whenever you see diameter, you're supposed to change that into radius. K? So one is double the other. So the pies will cancel. And I can say that a is r. And then this guy here is 2 r times v. So look what happens. I have, I have r squared. This 2 here becomes a 4. 4 r squared. So the r squares cancel, and you're left with v over 4. K? So if the radius becomes twice as big, then the speed will become 4 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 4 times greater, which means that the speed has to go down by a factor of 4. So the answer will be v over 4. Cool? These are pretty popular. Hopefully this made sense. Let's keep going.
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