19. Fluid Mechanics

Fluid Flow & Continuity Equation

# Continuity / Proportional Reasoning

Patrick Ford

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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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