Vehicle traffic and fluid flow

Traffic can be difficult to model mathematically, comprising as it does thousands of drivers in metal boxes making their own decisions and moving in a coordinated – or uncoordinated – fashion. But at the risk of over-simplifying things, it can be instructive to treat certain road conditions as a fluid.

Consider bottlenecks, where a blockage reduces multiple lanes of traffic down to one.

Diagram comparing free flowing traffic in one direction to a bottleneck in the other, which forces all the cars into one lane
Diagram of a road bottleneck: the rate of cars entering the section is the same in both directions, but the reduced flow around the roadworks forces the traffic into one lane, slowing cars to a crawl (Image by Smurrayinchester, via Wikimedia Commons)
In the diagram above, roadworks have forced all the cars from three lanes into one. The restriction of the single lane determines how many cars can pass through the entire section of road in any period of time.

Since cars can’t magically appear or disappear, they must be going into the blockage at the same rate – if you like, we can call this the conservation of cars. But because the three lanes mean there are three times as many cars going in, to satisfy the law of conservation of cars, they must be moving at one third the speed.

This may seem counter-intuitive, after all we normally assume that wider roads make traffic move faster. But you’ve probably observed the effect yourself when you encounter roadworks. It implies that widening roads won’t help if there’s some form of limiting factor, like an exit to a freeway. Adding more lanes to the diagram above will only cut the speed further.

This same principle applies to fluids, except it’s not conservation of cars, it’s more like conservation of mass. And the overall rule is called the continuity principle.

You can easily see the continuity principle in action in your home: for instance, water coming out of a tap accelerates under gravity, so when it’s moving faster the cross-sectional area has to reduce to keep the rate of flow the same. Which is why the stream narrows further from the tap.

Or for another example, when you put your thumb over the end of a hose you restrict the area it can flow through, and so the water moves faster – and you can squirt the water further.

But as a final interesting twist, there is a maximum speed you can reach by reducing the area like this: thanks to Bernouilli’s principle, increasing the speed of flow also reduces the pressure. And eventually the pressure gets so low that the water can actually boil at room temperature.

This is what sets the maximum speed: it’s known as choked flow. And this room temperature boiling, with bubbles of vapour forming and collapsing, is the cause of the characteristic hissing noise of taps.

And that’s what freeway construction and roadblocks have to do with noisy plumbing!

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