Engineers that work with fluids need a solid understanding of how they behave, and there’s one branch of fluid mechanics that plays a role in areas all across our lives: hydrostatics. Whether you’re designing a water tower for a city or you just want to understand how those upside-down pet bowls work, you’ve got to know how to relate the depth and pressure of a fluid.
I’ve put together this drawing of a few different containers filled with water. Try to think about if you put a pressure gauge at each red dot, which one would have the highest water pressure. Made a decision? The correct answer is the pressure at the bottom of each container is exactly the same, because they all have the same depth. In fact, for a fluid that’s not moving, or static, the volume or distribution of the water doesn’t matter - the only thing that affects the pressure is the depth. But why?
You may have heard of Bernoulli’s Principle, which basically says that all fluids have energy, and that energy can take one of three forms: kintetic, potential, or internal energy. Static fluids, or fluids that aren’t moving don’t have any kinetic energy so we’re left only with potential energy (which is related to the height of the fluid) and internal energy (which is related to the pressure). And this becomes really useful because any body of static fluid will have the same amount of energy no matter where you measure it. In a tank of water, at the surface of the water, the height is greatest, and the water pressure is zero. As you move down into the tank, the height decreases, but the water pressure goes up, so there’s no change in energy. At the bottom of the tank, we can say that the height is zero, so all of the fluid’s energy is in the form of pressure. Notice that in Bernoulli’s equation, there is no volume, total weight, mass, or distribution, it’s just height, pressure, and the unit weight of the fluid that relates the two.
Look closely and you’ll see that this relationship has applications all throughout our lives. Here’s an example: Any city has a grid of pipelines carrying fresh water to all of the buildings. Often called water mains, these pipes are under a significant amount of pressure. Imagine if you could tap into a water main with a vertical pipe. How high would the pipe have to go to keep from overflowing? We can calculate this using Bernoulli’s equation simplified for a static fluid. A typical pressure for a water main is 60 psi or around 4 bar. The static head equivalent, in other words, the height of a column of water with that same amount of energy, is just the pressure divided by the unit weight of water. For a typical water main, we find that the static head is about 140 feet or around 40 meters.
In fact, if you could do this at every point in a city and connect all the points, you’d have what we call a pressure plane - an imaginary surface almost like a comforting blanket, hovering above our heads representing the water pressure at any location. Actually we do have pipes exactly like this in almost every city. I’m talking about water towers. The height of the water surface in a water tower corresponds exactly to the pressure in the water mains. These are locations where the imaginary pressure plane has a real-life physical representation and this is why water towers are all usually around the same height.
This idea that hydrostatic pressure is only a product of depth isn’t completely intuitive. Let’s look back one more time at the tank illustrations. Intuitively you might assume that the pressure at the bottom of this tank is much higher. After all, it has a large volume of water presumably being held up, so to speak, by a very small column of water. And do you really think that a tiny tube with barely any fluid at all can increase the pressure of this entire tank by the same amount? Well, let’s build a tank and find out.
Here’s the setup. I’ve got an acrylic water tank with a balloon in it filled with air. I also have gauge on top of the tank so we can see the pressure. The tank is sitting on the ground outside my house and I am on the roof with a contraption of my own design - a broom handle with a pitcher duct-taped to the top. The pitcher has just a little bit of water in it, and the end of a tube is submerged in that water to keep any air bubbles from getting in. It runs down the roof to connects to the tank. This tube has a inner diameter of only ⅛” or about 3 millimeters, so the amount of water pushing down on the tank is miniscule, less than half a pint. As I walk up the roof and lift the pitcher higher into the air, watch what happens at the tank. The pressure goes up maxing out at 12 psi or about 0.8 bar. I put the balloon in the tank as an indicator to prove the the pressure did indeed go up. Looking at this tank, it’s hard to imagine that less than half a pint of water could affect it in any way, and yet such a small volume was still was able to shrink the balloon to almost half its size, just because I gave it some height. So why does this work?
Pressure is force divided by area, the force in this case being the weight of the fluid. If you shrink a column of water but keep it at the same height, you decrease the force, but you’ve also decreased the area by the same amount. It doesn’t matter how wide or skinny the column is. As long as it has the same height, it will always have the same pressure at the bottom. Being in a tank connected to a narrow tube of water like in the experiment, you would feel just as much pressure as if you swam to the same depth in the ocean.
For now let’s take our newfound knowledge of hydrostatics and apply it to a problem that has probably confounded all of us at one point or another: those watering bowls with an inverted tank. Why doesn’t the water come flowing out? If we use Bernoulli's theorem to analyze this, something’s not adding up. We have two water surfaces with no pressure, and yet they’re at different heights. How can the energy be the same on both sides? To answer that, we need to include the effect of another fluid in the system. Just like animals at the bottom of the ocean, we live underneath a sea of fluid as well: the atmosphere. If we include air pressure in our analysis, the math works out just right. On the drinking side, the surface of the water is exposed to air pressure, but inside the tank, the atmosphere can’t apply the same pressure. The top of the tank is a vacuum, it has a pressure lower than atmospheric. This difference in air pressure makes up for the discrepancy in energy that we noted earlier. If you expose the water surface inside the tank, by, for example, poking a hole, the atmosphere literally pushes the surface of the water back down.
But if it’s the atmosphere that’s pushing the water up higher into the tank, how far can it go? The atmosphere isn't infinite, so presumably there’s a point where it can't push any further - a point at which the column of water in the tank has the exact amount of energy as the column of atmosphere extending all the way into space. What if we build a taller cat bowl?
I’ve put together a few lengths of PVC pipe with a valve at the bottom and a transparent section at the top so we can see what’s happening. We hoisted this pipe to stand vertically and submerged the bottom into a bucket of water, similar to the drinking bowl you saw earlier. I filled up the pipe with water and put a cap on the top. If you know your weather, you might realize what we’ve created here: a barometer. Once the valve at the bottom is opened, this is essentially a balance beam weighing the column of atmosphere above us compared to a column of water. And it turns out that living at sea level on earth has the same absolute pressure as being 35 feet or 10 meters underwater. Theoretically, that’s the height of the column of water we should see in the pipe, but it’s not quite that simple. When we open up the valve at the bottom, watch what happens. Lots and lots of bubbles form in the water and the level drops.
There’s two things happening here. First, water has dissolved gases which come out of solution under a vacuum, creating some air pressure in the top and lowering the water level. Second, water is not a liquid at zero pressure, it’s a gas, so the water at the top of the pipe is actually boiling. Unfortunately I didn’t get the height of the pipe just right, so the water level settles just below the viewing window, and in hindsight it was pretty cavalier of me to just get such a small section of clear pipe, but it does reach a point where the vapor pressure plus the water column is equal to the atmospheric pressure outside - and you can tell because there’s no water coming out of the bucket at the bottom. There’s nothing tricky going on here. With a length of pipe or even vinyl tubing, and maybe a couple of googly eyes, you can boil room temperature water just by bringing it to the top of a parking garage.
But, you see why we don’t use water for barometers: besides the whole boiling part, this isn’t going to sit very nicely on the desk of your favorite meteorologist, which is a totally normal thing to have by the way. Instead we use mercury which is about 13 times as dense, so the column can be about one thirteenth of the height, and and even with the advent of more sophisticated ways to measure it, we still often use units of inches or millimeters of mercury to describe atmospheric pressure today: just another example of hydrostatics in our everyday lives.