### Physics of the Everyday

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 one of humanity's prized inventions.

# Toilets

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

• driven fluid flows,
• the hydrostatic pressure of standing water, and
• the siphon.

# Toilets

The first effect we'll study is how to drive fluid flows in the first place. Consider the pressures on the left- and right-hand sides of the fluid below. What would cause the water to flow left to right across the pipe?

# Toilets

Now we study the hydrostatic pressure. Suppose you are swimming $\SI{1}{\meter}$ underwater and then dive to $\SI{20}{\meter}.$

Will you feel a change in pressure (e.g. in your eardrum)?

# Toilets

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 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. This is known as the siphon effect.

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 pressure at the ends.

# Toilets

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.$\\[1.3em]$

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.

# Toilets

Suppose we slowly add water to the toilet bowl below — e.g. by pouring it in from a bucket. What will happen to the water level in the bowl?

# Toilets

Suppose we fill the bowl quickly so that water moves up and over the U-tube, forming a seal as shown below. What would happen next?

# Toilets

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 flows out under the pull of gravity.
• This fills the bowl quickly $($in about $\SI{5}{\second}),$ raising the water pressure in the bowl.
• This creates 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.
• 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.

We'll wrap up by considering some of the design choices in the toilet.

# Toilets

Up to now, the toilet we've described is the simplest manifestation of the jet-less siphon toilet, which was popular in the United States until the late $1990\text{s}.$ As they require a large volume of water $(3.5\text{ - }5\text{ L})$ to initiate a flush, they were phased out of use.

In their stead are so-called siphon jet toilets. Whereas jet-less toilets depend on the rising hydrostatic pressure of water in the bowl to create the siphon, the siphon jet toilet directs most of the water flow from the tank toward a small outlet just in front of the U-tube. This causes a very fast flow rate over the U-tube that can kick-start the siphon much faster than a jet-less toilet and uses much less water.

# Toilets

Suppose you build a toilet that is the same as your normal toilet but has a three-foot-tall bowl so that the height of the water column (blue in the diagram) that rushes in after a flush is higher as well.

Would this modification cause the toilet to clog more or less easily?

# Toilets

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.
• A siphon can transport water from high to low points.$\\[1.3em]$

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.

# Toilets

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