The Science of Everything Podcast - Episode 41: Flotation and Fluid Mechanics

Episode Date: December 25, 2012

A discussion of the behaviour of fluids, including an overview of the concepts of fluid pressure and Pascal’s Principle. I also discuss Archimedes’ principle of buoyancy and its application to why objects float, and Bernoulli’s Principle of the relationship between fluid speed and pressure, and how this can be applied (and misapplied) to explaining how aeroplanes fly. Recommended prerequisites: Episode 13 Newtonian Mechanics, Episode 27 Intermolecular Bonds and Phase Transitions.

You're listening to The Science of Everything podcast, episode 40. Boyancy and fluid dynamics. And I'm your host, James Fodor. In this episode, we're going to look at the behavior of fluids, particularly fluids in motion, hence fluid mechanics. We'll talk about fluid pressure, Pascal's principle, barometers, and we'll also look at Bernoulli's equation and some misapplications of Bernoulli's equation, and particularly how it's misused to explain how aeroplanes fly. We'll also look at buoyancy, that is why objects float and explain Archimedes principles. Recommended prerequisites for this episode, episode 13 on Newtonian Mechanics, and episode 27, intermolecular bonds and phase transitions.

All right, let's get started. First of all, some basic definitions. Volume refers to the amount of three-dimensional space that's taken up by a substance or an object. So volume is measured in something like meters cubed or liters. Density refers to the mass of an object divided by its volume, or in other words, it's the mass per unit volume. Pressure refers to the force divided by surface area, or, or the force per unit area, where the force is directed perpendicular to the area. So if I press a pin on a tabletop with my thumb, that will have a relatively high pressure,

because although my force is fairly small, it's just the force of a thumbbush, the area of contact between the table and the pin is very small. So my force is large, my area is small, so my force divided by area is large, hence my pressure is large. Similarly, if I have a very large mass confined in a very small space, my density will be very high. These concepts are fairly basic, and you probably know them, but I just wanted to make sure that they were clear

because they're going to be used frequently throughout the rest of the episode. Now, moving on from basic definitions to a recap on what we mean by solids and fluids and the difference between them. Now, we talked about this in episode 27, but just a brief recap. A solid is any substance that can hold its shape,

while anything that cannot hold its shape or that flows around other substances is referred to as a fluid. So fluids don't hold their shape. If you put them in a container, they will flow to take the shape of their container, whereas solids hold their shape. Liquids and gases are both fluids because they both flow.

Although we think of fluids as being liquids, gases are also fluids, because they don't hold their own shape. The big difference between gases and liquids is that gases are compressible, whereas liquids are not. And gases are also generally much less dense than liquids, so that means if you push on a gas, for example, if you put gas in a room and push all the sides of the room in, kind of like that scene in Star Wars where they're trapped in that trash compactor,

if you do that with a gas, the density of the gas will increase, because the mass is the same whereas you're reducing the volume. In other words, you're increasing the volume. the density, you're compressing the gas. If you do that with a liquid, it generally won't do very much. I mean, you'll be exerting a force on the liquid, but to increase the density of the liquid, you have to exert a very, very large force. Generally, it's not possible to compress a liquid. Okay, so that's what solids and fluids mean. So remember, whenever we're talking about fluids, it can be gases or liquids. Generally, it's easier to think of the liquid

in terms of understanding the model that we're thinking about, but it can be either. Okay, so now that we've defined solids and liquids, let's move on to talk about fluid pressure. Fluid pressure essentially refers to the force that the fluid exerts on its container, or even if it's not in a container, you can think about an imaginary parcel of fluid, you know, just an imaginary sphere of water, say, and the pressure of or in that sphere will be the force that's exerted on the water that's just around the outside of that sphere. So you can imagine that the sphere is itself surrounded by water, so its container is water. And so container doesn't have to be a little bit. container, it can be sort of an imaginary boundary. But in either case, pressure is simply the force that's applied perpendicularly to the container in which the fluid is located, or the hypothetical container in which is located. There's two main sources of this force. One is the thermal energy of the fluid, that is the random kinetic motions of the particles as they jiggle and bounce around owing to the energy that they have as a result of having your temperature, because temperature is just

average kinetic energy of the molecules in fluid, well in any substance, but including fluids. So as those molecules jiggle around and bounce around, they hit the sides of the container, and in doing so, they exert a force on the sides of the container. The component of that force that is perpendicular to the walls of the container, that is if you have a glancing blow on the side of the container, it exerts less of a perpendicular force than if you have a head-on collision. The perpendicular component of that force contributes to the outwards pressure on the sides or on the walls of the container.

So that's the thermal contribution to pressure. The second contribution to pressure is a gravitational contribution, essentially occurs because the earth or any massive object is constantly pulling down upon all of the molecules in the fluid. And so that's going to exert a downwards pressure. And so because of that, the fluid is going to exert a downwards force on the bottom of the container that they're in, or in other words on the fluid that's just underneath that they're sitting in. Gravitational force is obviously always acting downwards, so the gravitational component of pressure is always pushing downwards and not out sides like other forces of pressure can be.

So the pressure in a fluid will depend upon the temperature of the fluid because that will affect the average kinetic energy of the molecules within that fluid. But it will also depend upon the weight of the fluid located above whatever area you're talking about. So this is why, and this is an important principle, this explains why pressure under the ocean or under any body of water, in fact, increases with depth, because as you go deeper and deeper, more water is sitting above you, and therefore the pressure, say, on any given square centimetre of your skin or any given square centimetre of a submarine hull or whatever, the pressure on that given area will be equal to the total amount of weight of water that's located directly above that, that little bar or that little column of space,

directly above the area that we're interested in. As you go further down, the total weight of that column of water will increase, and therefore the total pressure increases. This principle does not just apply to the ocean or bodies of water, because it's a general property of fluid, so it also applies to the atmosphere, for example, or the air. So the total atmospheric pressure exerted on you or any other object at the surface of the earth is simply equal to the total weight of a pillar or column of air above that particular point. Well, the weight of that area, I should say, divided by the surface area of that column is referred to as atmospheric pressure, or one atmosphere. Because, as I mentioned before, gases are much less dense than liquids. Obviously, a given height of a column of water

is going to weigh a lot more than the same height of a column of air. But how much more you might ask? Well, one atmosphere of pressure, in other words, the weight of air that corresponds to the entirety of the Earth's atmosphere going up thousands of kilometres, is equal to the pressure increase that one experiences by diving to a depth of 10 meters in water. In other words, 10 meters high of water weighs the same as the entirety of Earth's atmosphere, assuming you're picking columns of the same circumference. Water is a heck of lot denser than air. This is also why there is a limit to how much air pumps can pump. They can only pump to roughly a depth of 10 meters, because beyond that, the weight of the atmosphere, which is what they're using to push up the water,

is insufficient to push the water more than 10 metres upwards, and so in order to keep pumping, you need to use compressed air or other mechanical means. This fact of the pressure increasing with depth has an important consequence for divers, because it can lead to a condition that's referred to as decompression sickness or informally the bends. Essentially this happens because as divers,

well, you know, dive deeper into the ocean, the pressure increases as a result of the greater volume of water that's above them, and at greater pressures, more nitrogen-differential. dissolves in the blood because essentially if you place the blood under greater pressure, more gas is able to be maintained in the water and you refer back to episode 31 on solutions and mixtures. We explain in more detail why this is the case. In a sense, what's happening is that the pressure from the water is just pushing, literally pushing the nitrogen into the blood, so more nitrogen dissolves in the blood. Now that in itself is not a problem. The trouble is when

divers ascend too rapidly from great depths, the pressure on their blood is reduced and the nitrogen comes out of solution. If that happens too rapidly, because they are ascending too rapidly, the nitrogen can form small bubbles, which can cause a whole bunch of problems. These are referred to as air embolisms. Essentially, they can cause blockages of blood. They can cause strokes and heart attacks at a worst-case scenario, but they can lead to a whole host of other smaller problems if one is not careful. And so basically, I mean, that can potentially be fatal if it's severe enough. So basically to avoid that divers have to ascend slowly or allow time for decompression once they've reached the surface. So that means they sit in a decompression chamber and the pressure is

gradually lowered so that over time they are gradually brought up to normal atmospheric pressure. That allows the time for the nitrogen two slowly come out of solution so that it doesn't form small bubbles or these air embolisms. Okay, so that's the basics of fluid pressure. I'm now going to talk about a somewhat more specific application of these general principles of air fluid pressure, which is Pascal's principle. Now Pascal's principle states that the pressure exerted anywhere in a confined incompressible fluid, so this mostly applies to liquids which are incompressible, is transmitted equally in all directions throughout the fluid,

such that the pressure ratio remains the same. So essentially what this is saying is that if you have a combined body of incompressible fluid, or in other words, a combined body of liquid, that's in contact with itself, so it's, you know, in one's place, and you push on one side of this fluid to increase the pressure in one area or in one part of the fluid,

that increase in pressure will be, not instantaneously, but effectively instantaneously transmitted all throughout the liquid. And so you can't have a localized increase in pressure in an incompressible fluid. Any increase in pressure from one side or one area will be transmitted equally in all directions throughout the fluid so that the pressure is the same throughout. A good demonstration of this is Pascal's barrel experiment, where essentially

what he did is he inserted a long, 10 metre long, according to my source, a vertical tube into a barrel filled with water. Obviously, if you push a metal tube into water, that's going to increase the pressure right underneath where you're shoving the tube in. What he found, however, was that when water was poured into the vertical tube, thereby even further increasing the pressure. So it was a hollow tube, and when he poured water in that, the weight of the water further increased the pressure immediately underneath the tube in the barrel. What happened, though, was that the increase in the pressure caused the barrel to burst, not just spring a leak at the particular place where the tube was located, the entire barrel burst, which is an indication that the pressure throughout the whole barrel was increasing, expectantly at the same moment, and therefore this demonstrated in Pascal's principle that the increase in pressure is transmitted equally throughout the entire fluid or the entire liquid.

Another way we can understand or see the same basic principle is that when liquid is in hydrostatic equilibrium, or in other words when it's been allowed to flow so that it's reached an equilibrium state, liquid always rises to the same height in all open regions of a container. That is, if you'll mention a long pipe, a long horizontal pipe that has vertical sections of pipe sticking up above it, and these vertical sections of pipe are different shapes,

so some might be sort of V-shaped, some might be like an upside-down V, some might be thicker and thinner, some might be of different heights, Pascal's principle states that in hydrostatic equilibrium, the shape and sides and anything like that of these vertical tubes doesn't matter. Water will always rise to the same height across the whole portion of the horizontal tube that we're talking about, as long as, of course, the water can flow throughout the entire body of this object. So the shape of the container doesn't matter, just the height matters. Really what this is saying in a really crude sense is that water flows downhill, but it's a little bit more specific than that.

Now, I mean, this might sound really obvious, but it's instructive to understand exactly why that happens. So think about what would happen if Pascal's principle did not apply. Imagine if we had two of those vertical tubes sticking out from the horizontal tube, one of which was much thinner and taller than the other. Imagine that the thin tall tube had a level of water that was higher than the other tube. If this was the case, then the pressure at the very bottom of the thin tall tube would be greater than that in the other tube, because we know the pressure is simply determined by the weight of the water that's above a given area,

and a given circle of area underneath the thin tube will have more water above it than in the thick tube. so the pressure at that underneath the thin tube must be greater than the thick tube. But if that was the case, then water would flow from the area of higher pressure to the area of low pressure. In other words, it would flow from underneath the thin tube to underneath the thick tube, and therefore the water level in the thin tube would fall, and the water level in the other tube would rise, and this would continue until the water levels in the two tubes were equal, thereby demonstrating Pascal's principle.

Pascal's principle can be used to explain the phenomenon of hydraulic lift, whereby water tanks with two pistons of different size can be used to raise very heavy lows. Essentially how this works is you have two pistons, which is just essentially like you can think of it as a vertical tube of water connected by a horizontal section. And the two pistons, and a piston is the tube plus a sort of a plunger that you push into the piston, which then pushes the liquid down. In a system that utilizes hydraulic lift, the two pistons will have different areas. One will have a much smaller circumference.

This is the first piston, and the second will have a much larger circumference. So you've got a small and a large piston. Remember that, according to Pascal's principle, the pressure of the water in both pistons, and in the whole system must be the same. So that means if we push down from one side, one piston, particularly if we push down the first piston, the small piston, then the second one must be pushed up as well, and the pressure increase must be transmitted across the whole system equally.

But remember, pressure is simply force divided by surface area. But if pressure has to be the same in the small thin piston and the wide fat piston, then the total force in the wide fat piston must be larger than in the small thin piston, because the total force is simply area times pressure. Pressure is the same, but area is much larger in the Y fat piston. So the total force in that second piston must be much larger if you make the surface area of the second piston much larger compared to the first piston. So that means that with a relatively small initial force, you can lift a very, you can exert a very large force using the second piston.

So you push down in the first piston with a relatively small force. That pressure change is transmitted and results in an upward force in the second piston, which is much, much larger than the initial downward force in the first piston. And so you can lift a very heavy load. This is used, for example, in mechanics shops to lift cars and other vehicles, because they can lift a very heavy weight of a car with only a relatively small initial force. But, of course, there's no such thing as a free lunch. Conservation of energy must still apply. So, in order to get this benefit of a much larger force from the second piston,

you can only move the object a much smaller distance. So, in other words, if you push the first piston down, say 10 metres, maybe you'll only get 10 centimetres of rise from the second piston, except the force that that applies will be much larger. and total energy is essentially force times distance. And so as long as that rate, as long as that total quantity, force times distance is the same for first and second pistons, then energy is conserved and you're okay. But what this hydraulic system allows you to do is essentially convert distance into force.

And this is very useful because like human muscles essentially, or mechanical means as well, are limited in the force we can apply, but they're not really limited in the distance we can apply it through. So it's relatively easy for us to apply a small force for a long distance. Very hard for us to, or even impossible for us to apply a large force for a small force. distance. But to lift something like a car, you might need a large force, but only for a small distance. And so hydraulic lift allows us to transform our ability to apply a small force across a long distance to apply a large force from small distance. And this is the broad principle of mechanical

work basically. This is how levers work as well and how pulleys and other systems work as well. They just allow you to transform distance into force, essentially. So that's a practical application of Pascal's principle. Another application of Pascal's principle, or related concepts at least, is that that of a barometer, which is simply a device that measures altitude or how far you are you up. And this is possible because if you can measure current air pressure and you know what air pressure is at sea level, then you can compare your current air pressure to pressure at sea level and therefore see how much further up in the atmosphere you are. This is how altometers work on aircraft and how barometers work and how you can tell

how fire you up a mountain and all those sorts of things. Although the air pressure will also be affected by weather patterns, but altitude has a dominating effect in most cases. Okay, so that's as much as we're going to say about pressure and Pascal's principle, which is an application of fluid pressure. Now we're going to move on to talk about buoyancy, which essentially explains how things float and why something's float and other things don't. Archimedes' principle states that a fluid exerts an upward buoyant force

on an immersed or floating object that is equal in magnitude to the weight of the fluid that is displaced by this floating or immersed object. So this might sound like a mouthful, but it's actually a fairly simple concept and very powerful if you think about it. In other words, whenever you place an object on the surface of or inside that is immersed in a fluid, so this is a liquid or gas, but we'll just think about it as a liquid from now on. Whenever you place an object on the surface of or inside a liquid, that object, well, obviously occupies a certain amount of volume, otherwise it doesn't exist,

because if you have zero volume, you don't exist. And so imagine if you take a ball that has, I don't know, 10 square centimeters of volume and you put it inside a liquid. Well, the space that's now occupied by that ball was previously being occupied by the water. And so where did the water go? Well, it was displaced by our ball. And this is a general principle. Whenever you immerse an object in water or any liquid,

you displace the fluid, and the amount of fluid you displaced is equal, it's just the same volume of fluid as the volume of the object they're putting in there. When you have an object that's floating on the surface of water, floating is actually just a special case of immersion, because, well, not exactly, but floating is essentially partial immersion.

You can only have a float when you at least displace some of the liquid on which you were floating, and so therefore the case is the same for a floating object. whatever volume of the object is below the surface of the water, that is, whatever portion of the volume is not floating, must be displacing liquid, and therefore the volume of that displaced liquid will be the same as the portion of the volume of the object that is floating.

So it's essentially the same thing as immersion, it's just only a fraction of the object is below the surface of the water. What Archimedes' principle says is that the weight of the volume of this displaced fluid, so how much volume of fluid you displaced, that must have a certain weight associated with it, which depends on its density. and mass. The weight of this displaced fluid is equal to the buoyancy force. The buoyancy force is just a force that is directed on the object upward out of the water. So it's pushing the object out of the water.

The size of this is equal to the weight of the displaced fluid. Now, the reason for this is essentially the pressure difference between the top and the bottom of the object, which is a result of the fact that fluid and, remember, fluid pressure increases with depth. Imagine an immersed object at 10 meters high, because this allows for a simple comparison. We know that in water, a depth of 10 meters corresponds to one atmosphere of pressure. So the top of the object, say at the very surface of the water, is at one atmosphere pressure because all it has above it is the atmosphere. 10 meters below that, the pressure is two atmospheres, that is the one atmosphere of pressure from the actual air and plus one atmosphere of pressure from a 10 meter depth. So the pressure difference across this 10

meter long object is one atmosphere. And that one atmosphere pressure difference between the top and the bottom of the object leads to an upward force pushing essentially from the high pressure to the low pressure. And this force is called the buoyancy force. Now, an object float if the total size of the, so that magnitude of the buoyancy force is exactly equal to the weight of the object. Because there are two offsetting forces here. One is the buoyancy force pushing upward, and the other is the weight of the object pushing downwards, which is simply caused by the mass of the object being attracted by gravity of the earth. So whichever those two forces is larger, then the buoyancy force is larger, then the object will sink. If the buoyancy force

is larger, then the object will float. If the two forces are exactly equal, which is pretty unlikely because, I mean, they have exactly equal. But if they were, then the object would have what we call neutral buoyancy. And this is what fish do, or at least approximately maintain roughly neutral buoyancy, by the use of a swim bladder whereby they can change the amount of air they hold within their bodies, basically, thereby changing their buoyancy. So, in terms of understanding how things float, you can think of it as this way. When you first place an object on the fluid, on the surface of the water, say,

it will only displace a very small amount of the fluid. And hence, the buoyancy force, which, remember, is equal to the weight of the displaced fluid, is only very small because the displaced fluid is very small. The weight of the object, however, while the weight of the object is constant, that doesn't depend upon how much it's immersed in the water. So the weight is constant, and let's say, relatively large, the buoyancy force is very small, so the weight force exceeds the buoyancy force and the object sinks. However, as the object continues to sink, more fluid is displaced,

and therefore the buoyancy force increases. And that continues to happen until either the entire object is submerged in the water, in which case the buoyancy force can't continue to increase beyond that point because there's no way the object can displace more water, because, well, the entire object is already in the water, so there's no more displacement to happen. If the weight force still exceeds the buoyancy force, even when the object is completely immersed, then the object will just keep sinking. If, however, you reach a point whereby the buoyancy force is equal to the weight force, then the object stops sinking, the forces are equalized, and the object will float. And whatever proportion of the water, sorry, whatever proportion of the object remains above the water, when this point of equilibrium is reached, will be the proportion of the object that, remains above the surface of the water when it's floating.

So using this basic principle, we can say that if an object is fully submerged and the buoyancy force still does not exceed the weight force, then the object must have more weight than the water that it's displaced, or more mass than the water that it's displaced. However, we know the volumes of the two objects, that is, the object and the water that it displaces are always the same, basically by definition. So if we take the mass and divide it by the volume, what we're left with is the density. So therefore, we can say that an object will only float. if it is less dense than the fluid in which it is placed.

And this kind of makes sense, but from Archimedes' principle, we can understand exactly why this is the case. It's just essentially a very simple mathematical procedure to show that. So this is why light objects float and heavy objects sink. It's because anything that is less dense than water will float on water, anything that's more dense than water will sink. Now remember, this density refers to average density across the entire volume of the object. That is why ships can float,

because obviously the density of iron and other things like that, of which the ship is made, is greater than water. what matters is the overall average density of the entire ship, and a large portion of the volume of ships is just air, essentially, empty space. So the average volume has to be less than the... Sorry, the average density has to be less than the density of water if they are to be able to float. And it's the same principle with the hot air balloon.

The average density of the hot air balloon must be less than the density of air, because otherwise they wouldn't be able to float in the air, which essentially what they're doing when they're flying. The way this is attained, by the way, is because hot air has a lower density than cold air. Might talk a little bit about more of this, and in a future episode, when we'll talk about flight. The great application of this principle of density is that of icebergs. So pretty much everyone knows that the overwhelming majority of an iceberg is actually below the surface of the water.

The actual figure is about 89%. It will depend on exactly, you know, the salt content of the ice and the salt content of the water and the temperature of the water, but around 89%. The reason for this is because, not coincidentally, ice, on average, has exactly 89% the density of cold water. So therefore, you need to get 89% of the volume of the ice under the surface. surface of the water in order for the ice to be able to displace a weight of water equal to the weight of the ice, of the total iceberg. If the density of water was exactly the same as the density of ice, then you would have to displace 100% of the volume of ice in order to displace the volume of ice in order to displace the same weights and therefore maintain neutral buoyancy.

But because ice is a little bit less dense than water, the iceberg needs to only sink 89% of its volume into the water in order to displace enough liquid in order to be able to float. Okay, so that's all I have to say about buoyancy. Now let's move on to yet another application of fluids and fluid mechanics, particularly this is an application of fluid mechanics, which is Benuli's equation, or Benuli's principle is also called. This is very frequently taught, but very poorly explained in physics textbooks and physics classrooms. In fact, I'd probably say it's one of the most poorly explained principles in all of physics, perhaps second only to the uncertainty principle.

But first, before I explain exactly what I mean by that, let's talk about what the principle is and what it says. Benoli's equation is simply a restatements of local conservation of energy, so it's nothing dramatically new, but it's a useful application. In its simplest form, Bernoules equation states that for an incompressible non-viscous fluid flow, there is a trade-off between the pressure of the fluid and the velocity at which the fluid's travelling. That is, pressure in a fluid decreases as its velocity increases, or alternatively as the velocity decreases, the pressure increases. Now remember, this only

applies for an incompressible fluid, so basically think about liquid, this wouldn't apply to gas because that is compressible, and it only applies to non-viscous flow. Viscosity, which is essentially the sticking force of a fluid. So honey is the canonical example of a viscous fluid. Honey is a fluid because it flows, but it flows very slowly because there's a lot of internal friction, essentially between the molecules in the fluid. That leads to a slowing of the rate of flow, and we call that viscosity. Water is not very viscous. It flows very rapidly. Honey is very viscous. Tarr is very viscous. So viscosity can lead to complexities in flow, and if you have a sufficiently high viscosity, laminar flow is much less likely. So,

and also leads to other problems. And so the Bernouli's equation really only applies to incompressible non-viscous laminar flow, so simple situations, but still useful in many situations. So given those restrictions on when it applies, there is a trade-off between pressure and velocity. So essentially what this is saying is if I put a bunch of flowing water in a pipe with a certain thickness, you know, the water will have a certain pressure and it'll have a certain velocity and will flow through the pipe. But then if the pipe narrows and passes through a thin section and then widens out again, what Bernoulli's principle says, or Bernoulli's equation says, is that the, as the water passes from the thick section of the pipe into the thin section of the pipe,

the velocity of the water will increase and the pressure will decrease. Why does this happen? Well, because for an incompressible non-viscous fluid flow, it must be the case that the same amount of water or the same volume of water is passing through any given section of pipe. Regardless of how fat the pipe is, I can't have more water passing through the thin section of pipe than I can have through the fat section of pipe, because then there'd be a sort of build-up of water, and that would be impossible because you can't have an increase in density of the water, because we've assumed it's impressible. Well, there's nowhere for the water to go, because we've assumed that it's trapped within the pipe. So it's simply impossible for that to happen by the assumptions we've made.

And this is a good assumption for something like water passing through a metal pipes. The water can't be compressed, it can't go anywhere else. And so the amount of water passes through any given section of the pipe, regardless of how fat that section is, must be the same. However, if that's the case, then for thin sections of pipe, the water must pass through more quickly, or in other words, its velocity must be higher. Otherwise, there's no way you could get the same amount passing through a unit of time. Because in order to have a given amount of water passing through a given area in a certain amount of time, you can, a given section of pipe in, say, a second, you can either increase the fatness of the pipe, or you can increase the

velocity of the water so that more water gets through. You can think about it like a corridor. If you want to get a given number of people through a corridor, you can either make the people walk more rapidly, which is equivalent to an increase in the velocity of the water, or you can make the corridor wider, which is equivalent to changing them with the pipe. Essentially what Bernoulli's principle says is that if you have a bunch of people walking down the corridor, which then narrows, and the number of people stays the same and their sort of average separation from each other stays the same, so they can't bunch up. If that holds, and the people can't stop or go somewhere else,

then people must walk more quickly through narrower sections of the corridor, and more slowly through wider sections of the corridor. That's not a perfect analogy, but that's essentially what Bernoulli's principle is saying. Now, we've said that in order to pass through the narrower section of pipe, the water must accelerate. And again, this doesn't just apply to water, but it's just a good example. Well, we know from one of Newton's laws that force equals mass times acceleration. So therefore, if we have an acceleration,

and we know we have got a constant mass, because the mass of the water doesn't change, There must have been a force exerted. The only way you can get an acceleration is for a force to be exerted. So where did the force come from? The force must be in the direction of the narrow region, because essentially the force is acting on the water, pushing from the fat pipe region to the narrow pipe region, accelerating the water as it travels from fat to thin.

And then the force will act in the opposite direction as the water exits the thin section of the pipe and returns to the fat section. Where does this force come from? Well, the force comes from the pressure difference between the wide and the narrow regions, and this is how we know that there must be a pressure difference. In order to get this force,

the only source of the force is going to be a pressure difference, that is a high pressure in the fat section of pipe and a low pressure in the thin section of pipe where the water is travelling more rapidly. Now, this is a little counter-intuitive. You might have thought the exact opposite that the pressure would be higher in narrow sections of pipe. That would be the case for something like a gas

if you were compressing the gas, but remember liquids are incompressible, and we're making the assumption that it's completely incompressible in this case. And so the thickness or width of the pipe doesn't matter in that sense. What does matter, though, is the speed at which the fluid is traveling.

Greater speed corresponds to a lower pressure, and that's where the low pressure comes from. It's not because the pipe is thinner per se. It's because the thinness of the pipe requires that the water travels faster, and that increase in velocity then leads to a reduction in pressure in the narrow region of pipe. Essentially what's happening, and remember before I said that this is an example, the Benuli equation is an example of a local conservation,

the local conservation of energy. Well, this is because essentially what's happening is the water that's flowing through the pipe has two basic sources of energy that we're interested in. It has others too, but only two that are relevant. One is the internal pressure or the forces inside the flowing liquid, and the second is the velocity or the kinetic energy that it carries as it travels. Now, we can convert one source of energy to the other, so we can convert kinetic energy to pressure or vice versa. In order to travel through the narrow section of pipe, at the same rate as water is traveling through the fat sections

of the pipe, we know the water has to accelerate. There's no other way it can happen. In order for that to happen, the kinetic energy of the water must increase. But if the kinetic energy has increased, the pressure, or pressure energy, if you want to think about it like that, must decrease. And so that's where the energy comes from. The kinetic energy of water in the thin section of pipe goes up, and so the pressure energy, or in other words, just the pressure of the water inside that thin section of the pipe must fall. Essentially, that's what's happening here.

There's just a transfer in energy between pressure and kinetic energy. And then when the water exits the thin section of pipe and returns the fat section, the transfer will reverse and the pressure will increase again and the velocity will decrease. There's no need for any outside intervention or anything like that. this just happens automatically in situations where you have a changing width of the diameter of a pipe or another enclosure that a fluid can travel through. Now, you remember I said before, the Bernoulli's principle or Bernoulli's equation is one of the most poorly explained concepts in all of physics. Well, now that I've explained the principle, reasonably properly, at least to a basic level,

I'm going to explain how it's misapplied or misexplained. Now, a common classroom example, and this has even done in high schools and universities, and talked about in textbooks, including some of the physics textbooks that I own, will say something like, if you hold a piece of paper out in front of you, so that it's sort of flopping down, and then blow over the surface, the piece of paper will rise so that it's sort of horizontal, so that is parallel to the ground. And the stated reason for this is because when you blow over the top of the paper,

you're accelerating the air, so the air is traveling faster. By Bernoulli's principle, this leads to a reduction in the pressure above the paper, and therefore the pressure difference between the top and the bottom of the paper leads to the paper being pushed up horizontally. This is completely wrong. Now, it is true that when you hold a piece of paper like that and blow up to it, yes, it will rise up to the horizontal, or at least to some degree, that does happen. So it's empirically true, but the reason for it is not because of Bernoulli's principle. Exactly why the paper does that seems to be actually slightly complicated, uh, it depends. It depends upon precisely what principles or what aspects of what aspects or what principles or what aspects of what aspects or what principles or what aspects of what we're going to look at. Because there's a lot of complicated physics that come into play when you start talking about fluid flow, especially if it's not laminoroughough in that case. The reason is because Benuli's principle says nothing about the static pressure of free air as its speed changes. That is, Benulli's principle is only talking about the pressure of a fluid

that changes simply as a result in the change essentially of the container that it's traveling through. If you exert an external force on the flowing liquid in order to change its velocity, then that does not necessarily, or there's no reason that that would lead to a change in pressure. In other words, remember my pipe with the fat and the thin and then the fat sections, if I just took a big plunger and pushed one end of that pipe, one end of the water in that pipe, so that I was exerting a force on all of the liquid, this would increase the velocity of all of the flowing liquid

throughout the entire section of pipe without really changing the pressure at all. Or it may change the pressure to some extent if velocity is impeded, but there's no reason that by increasing the velocity there, I would have to reduce the pressure in the entire pipe. But normally applies as sort of a, when there's no external factors acting on. I mean, it can apply when there are external forces acting on the system,

but it does not need external forces acting on the system in order to be applied. So if I accelerate a bunch of moving air, that does not necessarily reduce the pressure in that air. The pressure in that air may be falling for other reasons, but my simply exerting a force on it is not going to, if so facto, that is, because of the fact that I'm exerting a force on it, it's not going to reduce the pressure of that air. That's a completely misunderstanding of a Bernoulli's principle.

So, in other words, the simplistic way that you can think about that is, If a parcel of air or fluid accelerates or decelerates by itself because of the shape of the container, then the pressure will change. If the velocity of the flowing fluid changes because of some external force acting on it, like me blowing, for example, then there's no reason the pressure needs to change. And now this leads me to the biggest misapplication of Bernoulli's principle around, which you may have heard of before. You almost certainly, if you've ever done any science ever, have been taught this, you may or may not know that it's wrong. And that is the application of Bernoulli's principle to how aeroplanes fly. The traditional explanation is that you've got a wing which is shaped so that the top is curved more than the bottom. In other words, you can think about it as sort of like a hemisphere.

The top is curved and the bottom is mostly flat. And this means that air molecules that are flowing along the wing sort of get separated. The ones that travel over the top of the wing have a longer distance to travel than the particles that flow below the wing. However, both the particles, the air molecules that travel above and below the wing must meet up again at the end of the wing. In order for that to happen, the particles that travel above the wing must be travelling faster. In other words, they must have an increase in velocity, because otherwise they wouldn't be able to travel a greater distance, because a curve in the same time, because they have to meet up with the bottom particles at the end. So, in other words, the shape of the wing causes the particles traveling above the wing to travel faster than the particles traveling below the wing.

And by Bernoulli's principle, when air particles travel faster, or when they increase in speed, their pressure falls, or the pressure of that air falls. And therefore, the pressure above the wing is lower than the pressure below the wing. and this leads to an upward force pushing from the high to the low pressure, an upward force acting on the wing which results in lift. So this explanation that I've just given is, to put it bluntly, wrong, wrong, wrong. It's completely wrong. There are a few aspects of it that are sort of correct, but for the most part it's very wrong and very misleading.

The one thing that is true is that air does travel fast across the top of the wing and across the bottom, and the resultant pressure difference does contribute to lift, but that is the extent to which Bernouza principle applies to explain lift. The crucial thing that is wrong about the explanation that I just gave is the equal transit time aspect of it, that is that the molecules that travel above the wing have to meet up with the molecules that travel beneath the wing. That is simply wrong. It's empirically false that that happens. There are ways that this can be tested in an experiment, and what is found is that the molecules that are traveling on top of the wing

do in fact travel faster than the molecules that travel below the wing. However, they travel even faster than would be predicted by equal transit time theory. That is, the molecules above the wing actually get ahead of the molecules below the wing. a wing. They never meet up again. So there's no equal transit time. That's completely wrong. Another thing that's wrong about this explanation is that it's not always the case that the top of a wing is more curved than the bottom of the wing. Often

this is the case, but it needn't necessarily be the case. Some wings are essentially just flat and they fly perfectly fine. Some wings are actually curved on the bottom and they fly fine as well. And planes can also fly perfectly fine upside down, which would be inconsistent with this equal transit time Bernoulli's principle explanation of lift. So, you might be thinking, how do aeroplanes fly?

and the answer is, we don't know exactly. Okay, that's not exactly true. It's more that we know how airplanes fly, but exactly, we know all the physics that is involved with airplanes flying. However, exactly how important different physical laws and principles are in any given situation is open to a bit of interpretation. The only way to determine exactly is to conduct very precise experiments and computational models, which are, as I said, expensive and difficult to carry out,

and also depend upon the precise parameters of the situation you've set up, like exactly the shape of the wing and exactly the... temperature and density of the air and so on. So there's no one simple answer that you can give as saying this is how airplanes fly everywhere and always, and this is the single principle that explains it. There's a wide variety of physical laws that are at play. All applications of Newton's laws, of course, it's not like there's any new physics here. It's simply that when you have fluid flow that is non-lamina, that is, it gets chaotic. The physics get very complicated, even though you're just applying simple rules, the behavior of this system can

become very complicated and that can lead to complexities in explaining exactly what's going on. My preferred explanation to how airplanes fly, which is not completely correct because, as I said, there's more complicated physics going on, but it's a very good first-order approximation, and it's much less misleading than the Equal Transit Time Benoulli principle of explanation, is simply that the wing of an aircraft is shaped such that it angles downwards when you're looking at the airplane from side on. In other words, as the airplane flies forwards, the wing is angled downwards, in other words, downwards from front to end, so that the The air that's travelling across the wing, both above and below, is angled downward.

It's pushed downwards by the downward angling of the wing. The fact that the plane, by the wing, is pushing air downwards, leads to an equal and opposite force of the air pushing the plane upwards, and this upwards force on the plane by the air is called lift. So essentially, the lift for an airplane is generated by the angle of the wing, pushing air downwards, and leading to therefore the plane in turn being pushed upwards. This also explains why airplanes have to move forwards in order to fly. They can't just hover, because in order to push that air downwards,

you need to have some forward momentum or forward velocity. Without that forward velocity, there's no air passing over the wing, and so there's no upward force. And in fact, the faster you fly, within reason, the larger the lift force will be, which is why airplanes have to have a long runway because they have to meet a certain minimum speed before the lift force is sufficiently large

to overcome the weight force, the downward weight force, on the airplane. And this is sort of known as the Newton's law explanation of lift. It's just an application of equal and opposite forces, basically. It's not perfect, because there are aspects of airplane flight that this doesn't explain. it also ignores the fact that there is actually a pressure difference on both sides of the wing because of the difference in air travel velocities. So Benulli's principle does come into play.

However, the reason I like this Newton's Law explanation is because it's simple, and it is one of the very important, not the only important, but one of the very important physical laws that's relevant to how airplanes fly. And most importantly, it is a hell of a lot more accurate than the Equal Transit Time Bernoulli's Principle explanation, which is just so very wrong, wrong. Now, be warned, physics professors and physics textbooks still teach this. I don't really know why, that if you look it up online and there are articles written on this, technical articles, Wikipedia has excellent resources, the equal transit time theory is simply wrong and that explanation of Lyft is inaccurate.

So don't be afraid to challenge a physics professor or someone like that if they start to teach you this nonsense. Tell them to look it up and they'll find that in fact you are correct. Okay, now that that rant is over, the episode is also coming to a conclusion. That's all I wanted to say about fluid mechanics. Hopefully you enjoyed the episode. If so, please log on to our Facebook page. Just go to Facebook and type in The Science of Everything podcast, and you'll find our page where you can like the page,

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