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. Alright? We're going to discuss a really important equation, and then I'll show you an example. Alright? So continuity basically says that because ideal fluids are incompressible because their densities can't change, they can't get more or less dense, and they can't get squeezed, then the volume flow rates of a liquid have 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. We can say that remember Q, which is given as ΔV over ΔT . We saw that in the last couple of videos. We can also rewrite this in terms of area times speed. All these equations are sort of equal to each other. That has to remain constant. Alright, 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 pipe or you have some water that's flowing through this pipe over here and it's got a cross-sectional area A1, 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, somewhat of volume that's flowing through this pipe. Let's just go ahead and call this 10 liters, just to give it some numbers. Right? What continuity says is that if you have 10 liters of water that's flowing through this pipe every second, then later on if the pipe changes its area, its cross-sectional area, you still have to have 10 liters that are traveling through the pipe. You can't have 5 liters because then where did all the extra water go. And you also can't have 20 liters because then all of a sudden you've 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. Okay? Now what we can see here is that in this pipe, the cross-sectional area has gone down. This cross-sectional area over here, A2, 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 Q 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. Alright. So 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. Alright. And here's probably the most important equation that you need to know. The A 1 × V 1 = A 2 × V 2 . Okay. This is basically the equation that you're going to use when you talk about continuity. Alright. The last thing I want to mention here is that pipes are usually cylindrical, which means that their cross-sectional areas are going to be or π r 2 . Alright? 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 stick your thumb over here. This is going to be your thumb like this, 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 red. So you've got an '..
19. Fluid Mechanics
Fluid Flow & Continuity Equation
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