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Toilets are everywhere, in our parks, our schools, and even in our homes. Whether or not you'd like to admit it, toilets (hopefully) play a big part in your life. But how do they work?

Though it may seem like magic, the toilet's disappearing act is straightforward, based on elementary principles of fluid mechanics. Here we'll develop the fluid principles we need, and apply them to understand humanity's most widely used invention.

The toilet operates via a chain of effects in fluid mechanics. The crucial ideas we'll need to explain it are

- the hydrostatic pressure of standing water
- driven fluid flows
- the siphon.

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The siphon is an old technology used to transport water from higher to lower locations, and is commonly used to clean fish tanks, transport water for irrigation, and sometimes to steal gasoline. If a tube is completely filled with water (i.e. it has no air gaps), then hydrostatic pressure (due to gravity) at the high end will drive water to the low end.

Importantly, it does not matter if the tube goes up before it goes down to the lower end, and all that matters for the siphon effect is the relative height of the ends.

We've now established three important principles:

- Fluid pressure increases with depth in any column of water.
- Fluid flows when it is subject to a difference in pressure.
- A siphon can transport water from high to low points.

Simple as they are, these ideas are at the core of toilet functionality.

Now that we have the theory laid out, let's build up the mechanics of the flush.

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We now have everything we need to understand the function of the common household toilet.

- When we flush, the tank on the back releases water that flow outs under the pull of gravity.
- This fills the bowl quickly (in about \(\SI{5}{\second}\)), raising the water pressure in the bowl.
- This creates to a large difference in pressure between the water in the U-tube, and the water in the bowl, driving water up and over the U-tube, into the sewer pipe.
- As we saw, this forms an airtight seal between the water and the tube, which turns the bowl:U-tube into a siphon.
- The siphoning then drains the entire contents of the bowl into the sewer pipe, after which the siphon seal breaks and the bowl is empty.

Now, we'll wrap up with some consideration of design choices in the toilet.

Does the height of the water tank have a significant effect on the function of the toilet?

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In this quiz, we laid out three basic principles of fluid mechanics and applied them to understand the workings of the toilet.

- Fluid pressure increases with the height of any fluid column.
- Fluids flow down pressure gradients.

The second principle is a special case of what is known as Bernoulli's principle, which relates the pressure, potential, and kinetic energy of a fluid in pipes. In the general case, these mechanics are governed by the Navier-Stokes equations.

Fluid mechanics is one of the more universally applicable aspects of physics, and applies to much more than just toilets. Continuing on with the principles touched on here will lead us to understand global weather cycles, airplane turbulence, and even the course of hurricanes.

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