Fluid flow

0:00- [Instructor] When you suck on a straw which is dipped in water, the water rises up. Pretty simple, right? But here's the question, what makes the water rise up? What forces it to go up? If you're thinking suction, then we'll have to think again, okay? And what about giant pipes that we have in our cities and the ones that come to our houses, what makes the water flow there?

0:20Is it the same forces that we're dealing with over here or something different? That's what we're gonna figure out in this video. So let's begin. We'll start with the most fundamental question, what makes a fluid flow? Well, we know from Newton's laws, if there's an unbalanced force acting on an object, it'll accelerate.

0:37Well, something very similar is gonna happen over here. Instead of forces, we're gonna deal with pressure, if there's an unbalanced pressure, or in other words, if there's a difference in the pressure between two points and a fluid, that fluid is going to flow.

0:49That's what you need, a difference in the pressure. So let's use this insight and see if we can answer our original questions. So let's go back to sucking that water on the straw, okay? And let's look at this from the side view. Let's concentrate on the column of the water that is inside the straw over here, only on that particular column of water.

1:09And think about what are the forces that are acting on it. Well, first of all, we know the air is pushing from the top. So there is this atmospheric pressure. We know that this column of liquid has its own weight, gravity's pulling down on it. So there's a force of gravity acting on it.

1:24And we know that there is a pressure from the bottom. We'll just call that is Pb. And I know all these forces are balanced because this water column is in equilibrium. It's neither accelerating up nor down. So how do I balance these forces? Well, the forces from the bottom should exactly equal the forces from the top.

1:41What's the force from the bottom? Well, we know the pressure from the bottom, well, if I consider the area of the cross-section to be just A, then the force from the bottom is pressure times area. What's the force from the top? Well, is the gravitational force plus this pressure times area.

1:55So we can write down an equation. We could say that the force from the bottom, which is the pressure times area from the bottom, that should equal the force of gravity plus the pressure times area from the top. Does that make sense? Okay, but we know what the force of gravity is is just mass of this column of liquid times g.

2:13And we know mass is just the density times the volume. So it'll be the density of this fluid, which is water, times the volume of this particular column. So we can just write as density times volume. And finally, what's the volume of this column of water? Well, it's a cylinder.

2:30So the volume of a cylinder is just the area times the height. So if we plug that in, we get our final expression. This is now the equilibrium condition. This is the case when there are no net forces acting on this column of water, it's in equilibrium. But now let's see what happens when you suck on that straw.

2:48Well, when you do that, some of the air molecules are gone from from the straw. And so the air pressure over here drops. So the air pressure reduces, okay? but what about this pressure from the bottom? Well, that stays the same, because that depends upon the depth and the atmospheric pressure outside, which is the same.

3:07Only the pressure inside the straw has reduced, which means, look, the left hand side side stays the same, but the right hand side has reduced and therefore this is no longer equal, now the force from the bottom is larger than the total force from the top and therefore there will be a net force acting upwards pushing the liquid up.

3:30So you can see what causes the net force acting upwards, the difference in the pressure, that's what's causing it, and that's what's going to push the liquid up. There is no suction pull, but it's actually being pushed from the bottom because the pressure from the top reduced.

3:45That is how the liquid comes up. But now the next question could be, well, how high will it rise? Well, again, think about it as the height increases this value starts increasing, right? So at some height, let's call it a h prime for example, some new height, the sum will, again, be equal to the left hand side.

4:04And when that happens, another equilibrium state is reached and that's when the water stops rising. Again, long story short, what caused the water to rise up, it's not a suction power, there is no such thing as suction pull, it's due to the difference of the pressure.

4:17Because the pressure from the top reduce the pressure from the bottom, that is the one that pushed the liquid up. All right, onto the next question. What makes the water flow in these pipes into our houses, into our taps, for example? Is it the same phenomena?

4:31Is the same idea? Or is it something different? Why don't you pause the video and think about this. All right, so usually we'll have a giant water tower. So the whole idea is that because of this huge height, the pressure of the water here would be very high. It's gonna be the same equation as you saw before, the atmospheric pressure plus rho gh.

4:52Now, because h is big, this number is pretty big. And so the pressure at the bottom is very high. And that makes sense, right? All of that water is pushing onto it, providing a lot of pressure. All right, now what about the pressure in your houses? Let's say when you open the tap, for example, what is the pressure over here?

5:12Well, when you open the tap over here, this is open to the atmosphere. So the pressure over here is the atmospheric pressure. Ooh, can you see? The pressure here is much larger than the pressure here. And therefore, that difference in the pressure will push the water into the tap over here.

5:31I mean, there's water everywhere, but it just pushes, now the water starts moving and that's how the water flows out. So the idea is exactly the same as before, is the difference in the pressure that pushes the water. Amazing, right? All right, so the final question for us now is what decides the speed at which the water flows?

5:50How do we figure that out? Well, let's see. Let's zoom in over here, and let's consider a couple of cross-sectional area. Let's consider a cross-section area over here as A1 and let's consider the cross-sectional area over here as A2. Our goal is to think about what is the connection between the speed at which the water is flowing here and the water is flowing over here.

6:08How do we think about that? Now, if you model the water as an incompressible fluid, then we could say that the volume of the water that's flowing in over here per second, that's what we call the volume flow, how much water is flowing in per second, that should equal the volume of the water that's flowing out here per second.

6:29Does that make sense? Think about it, if there is five ml of water flowing in over here every second, well, then 5 ml of water should come out every second. If less comes out, then there is some water that's getting compressed over here, right? So that can't be true.

6:41It has to be exactly the same if we model it to be incompressible, which is a pretty good model, okay? So now let's see if we can figure out what the volume flow rate over here is. How do we figure out the volume flow rate? Well, there are multiple ways to think about it.

6:56Here's how I think about it. Let's assume that the speed of this water is flowing in, I don't know, let's just call it as V1 and the speed of the water is flowing out as V2, right? Now consider some water molecules that are right over here. Now, if I wait for one second, then in one second these water molecules will move to the right because the whole thing is moving to the right.

7:17But how much does it move to the right in one second? Well, its speed is V1, that means it moves, it travels V1 meters per second. So if I wait for exactly one second, these molecules would've traveled the distance of V1 meters. Make sense? Okay, now, there are some molecules over here before that would've also traveled the exact same distance and they would've come over here.

7:43There were some molecules to the left of that, they would've come over here. There are some molecules over here, they would've come over here. And there were some molecules over here, they would have eventually come over here. So this means in that one second that we waited for, look, this much fluid, this much water went through this cross-section.

8:01So this is the volume of the water that went through in one second. That's the volume flow rate, isn't it? So I just need to know what the volume of this is, that would be the volume of the water that flew in in one second. So what's the volume of this? Well, we know volume of a cylinder is just area times the height.

8:21Well, area is A1, height is V1. So the volume flow rate moving into it is A1V1. Similarly, the water is flowing out over here, the idea is exactly the same. And therefore, here the volume flow rate flowing out would be A2V2. And since these need to be equal, there we have it.

8:44A1V1 should exactly equal A2V2. We call this the continuity equation. And let's see what that tells us. This tells us that as water flows, if the cross-sectional area reduces then since the product needs to be the same, the velocity should increase. Which means in this particular example, look, as the water goes from here to here, the cross-sectional area has reduced so the water must go out faster.

9:09So the water is moving in slower and it's coming out faster. And again, intuitively that makes sense, right? We want the same volume of water. If 5 ml is going in every second, 5 ml must come out. But since the area is smaller, well, the water molecules have to move faster so that the total volume stays the same.

9:26Because if the area is smaller, well, then it needs to move faster to have traveled a larger distance, right? So that makes sense, isn't it? This is how we are using continuity equation. If we know the speed of water on one side, we can figure out the speed of water somewhere else in a pipe, for example.

9:40This is also the reason why, for example, when you pinch a hose, for example, the water comes out faster, is the same idea. When you pinch the hose, the area of the opening reduces, but the volume flow rate has to stay the same so the water comes out faster. Beautiful, right?

9:56And the same idea holds if you have slightly more complicated pipes, like for example, if you have like two incoming waters join and you have an outgoing water over here, for example, or fluid over here, it works for any fluid, okay? The same idea holds true, right?

10:10The volume flow rate in, that's together, that's A1V1 plus A2V2 should exactly the volume flow rate out, which is equal to A3V3.