The Universal laws behind growth patterns, or what Tetris can teach us about coffee stains

The morning after a big snowstorm swept through the US northeast, I sat in my car, ready to brave hazardous road conditions and drive to the local coffee shop. My home in New Jersey was outside of the storm’s central path, so instead of piles of snow, we were greeted with a delightful wintry mix of sleet and freezing rain. And sitting in my car, I couldn’t help but be mesmerized by these strange patterns of ice particles forming on my windshield. Here’s what I saw:

As I watched this miniature world self-assemble on my windshield like an alien landscape, I wondered about the physics behind these patterns. I learned later that these patterns of ice are related to a rich and very active current area of research in math and physics known as universality. The key mathematical principles that belie these intricate patterns lead us to some unexpected places, such as coffee rings, growth patterns in bacterial colonies, and the wake of a flame as it burns through cigarette paper.

Let’s start with a simple example. Imagine a game similar to Tetris, but where you only have one kind of block – a 1 x 1 square. These identical blocks fall at random, like raindrops. Here’s a question for you. What pattern of blocks would you expect to see building up at the bottom of the screen? Continue reading The Universal laws behind growth patterns, or what Tetris can teach us about coffee stains

Can we build a more efficient airplane? Not really, says physics.

Update (13 October):  I emailed David MacKay to get his opinion on some of the critical comments responding to this blog post. David is a physicist at Cambridge University, author of the book ‘Sustainable Energy – Without the Hot Air’, and is the chief scientific adviser to the UK Department of Energy and Climate Change. You can read his response in the comments below. There’s also a interesting discussion of this post over at hacker news.

Boeing recently launched a new line of aircraft, the 787 Dreamliner, that they claim uses 20% less fuel than existing, similarly sized planes.

How did they pull off this sizeable bump in fuel efficiency? And can you always build a more fuel-efficient aircraft? Imagine a hypothetical news story, where a rival company came up with a new type of airplane that used half the fuel of its current day counterparts. Should you believe their claim?

More generally, do the laws of physics impose any limits on the efficiency of flight? The answer, it turns out, is yes.

Jet Man, by Ben Heine

There’s something about flying that doesn’t sit well with us. If we never saw a bird fly, it may never have occurred to us to build flying machines of our own.

Here’s where I think this sense of unease comes from. It takes stuff to support stuff. Everyday objects fall unless other things get in their way. Take the floor away, and you’ll plummet to your doom – the air below your feet isn’t going to do much for you. We move through air so effortlessly, that we barely notice it’s there. So what keeps a plane up? There doesn’t seem to be enough ‘stuff’ there to hold up a bird, let alone a Boeing aircraft weighing up to 500,000 pounds. To put that last number in context, its more than the weight of an adult blue whale!

Why is it that planes fly and whales typically don’t? The answer is easy to state, but its consequences are rather surprising. Planes fly by throwing air down. That’s basically it. It’s an important point, so I’ll say it again. Planes fly by throwing air down.

As a plane hurtles through the air, it carves out a tube of air, much of which is deflected downwards by the wings. Throw down enough air fast enough, and you can stay afloat, just as the downwards thrust of a rocket pushes it up. The key is that you have to throw down a lot of air (like a glider or an albatross), or throw it down really fast (like a helicopter or a hummingbird).

A physicist’s two-step guide to flight (it’s simple, really!)

Let’s make this idea more quantitative. Following David MacKay’s wonderful book on Sustainable Energy, I’m going to build a toy model of flight. A good model should give you a lot of bang for the buck. The means being able to predict relevant quantities about the real world while making a minimum of assumptions.

Toy models gone wrong. By Randall Munroe at XKCD.

Step 1: Sweep out a tube of air

As a plane moves, it carves out a tube of air. This air was stationary, minding its own business, until the airplane rammed into it. This costs energy, for the same reason your car’s fuel efficiency drops when you speed up on the highway. Your car has to shove air out of its way.

Exactly how much energy does this cost? You might remember from high school physics that it takes an amount of energy equal to 1/2 m v^2 to bring stuff with mass m up to a speed v.

In our case, we have

There’s still this mysterious factor of the mass of the air tube. To work this out, we can use a favorite trick in the toolbox of a physicist – unit cancellation. We can re-write the humble kilogram as a seemingly complicated product of terms.

What we’ve done here is to express an unknown mass of air in terms of other quantities that we do know. Each of these terms makes sense. Air that’s more dense will weigh more. A fatter plane (larger cross-sectional area) sweeps out more air, as does a faster plane. We’ve arrived at a meaningful result, just by playing around with units. In the words of Randall Munroe, unit cancellation is weird.

Put these two ideas together and here’s what you find:

Here’s a graph of what that looks like.

If you’re with me so far, we just found that for a plane to plow through air, it has to expend an amount of energy proportional to the speed of the plane to third power. (The extra factor of v comes from the fact that faster planes sweep out a larger mass of air.) If you want to go twice as fast, you need to work 8 times as hard to shove air out of your way.

We’ve arrived at a general rule about the physics of drag. This holds true for a car on the highway, or for a swimmer or cyclist in a race. It’s why drag racing cars get only about 0.05 miles to a gallon! If we want to reduce overall energy consumption by cars, one option is to lower the speed limits on highways.

What does this mean for our toy plane? It would seem that the slower the plane, the higher its efficiency. So are airplane speed limits also in order? Absolutely not! To see why, read on to the second half the story..

Step 2: Throw the air down

In order to fly, a plane must throw air downwards. This generates the lift that a plane needs to stay up. It turns out that slower planes have to throw air harder to stay afloat. That’s why slow moving hummingbirds and pigeons have to flap their wings frenetically. It’s also why planes extend flaps while landing – they’re not throwing the air fast enough, so they compensate by throwing more of it.

More precisely, for a plane to stay afloat, the speed of the air jettisoned downwards must be inversely proportional to the speed of the plane. (You can take my word for this, although if you want to see where it comes from, take a look at David MacKay’s book.)

So we can now work out the second part of the puzzle. How much energy does it take to throw air down? As before, this is given by

Just as we did in the first step, let’s express things in terms of the speed of the plane.

In words, the energy spent in generating lift is inversely proportional to the speed of the plane. Here’s what this looks like on a graph.

You can see from the plot that, as far as lift is concerned, slower flight is less efficient than faster flight, because you have to work harder in throwing air downwards.

There’s a lot to chew on here. To summarize, we’ve discovered that in making a machine fly, you have to spend energy (really fuel) in two ways.

  1. Drag: You need to spend fuel to push air away. This keeps you from slowing down.
  2. Lift: You need to spend fuel to throw air down. This is what keeps the plane afloat.

The total fuel consumption is the sum of these two parts.