Fluid Flow & Continuity Equation

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 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 double 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 diameter of B will be twice that to D and we want to know what is the volume, what is the 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 the D 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 the area increases which it does here the speed has decreased so that the products A V stays the same, okay?

So, one way to think about, this is, 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, it's not going to be the same, it's not going to be faster. So, it's now down to whether it's V over 4 or V over 2 and what you can do is you can just write, you can write A1, V1 equals A2 V2, 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 VB. So, VB is the first area 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 square divided by pi, r squared times the first velocity which is V, okay? Now, I don't have the radius, 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 going to call this r and this is going to be 2r, okay? So, if the diameter doubles the radius doubles and all these questions whenever you have diameter, pretty much in all of physics whenever you see diameters are supposed to change that into radius, okay? So, one is double the other so the pi's will cancel and I can say that A is r and then this guy here is 2r times V. So, look what happens, I have, I have r square, this 2 here becomes a 4, 4r squared so the rs squares cancel and you're left with V over 4, okay? 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 makes sense, let's keep going.