# If you know what you are doing you get the same result from a 1940s differential analyser and a modern computer running Python

Above is an aircraft wing which has picked up ice on the leading edge of its wing. This is of course is not ideal, especially for an aircraft in the air. You don’t want large amounts of ice forming along the leading edge of the wing in flight, it will add weight and make the wing less able to generate lift. I think there were particular worries about this during the Second World War, possibly because planes were flying higher and faster as the war drove rapid advances in aircraft design and performance. So the United States Army Air Force turned to the dream team of a Nobel-prize winner, Irving Langmuir, and the first woman to obtain a PhD in physics from the University of Cambridge, Katherine Blodgett. They worked to understand the following problem.

Think about an airplane flying through a very cold cloud of tiny water droplets. If these droplets hit the wing they freeze into ice. Which is bad. So how many droplets hit the wing? The air flows over the wing, but the inertia of the droplets can cause a droplet not to follow the air over the wing, but to carry straight on and hit the leading edge of the wing. This inertia is quantified by what is called the Stokes number, which here is

```Stokes number = 106 U d2 / t
```

where U is the speed of the air of over the wing, d is the diameter of the water droplets, and t is the thickness of the wing. World War Two warplanes flew at speeds of several hundreds of miles per hour, which is about 100 m/s – the above formula needs the quantities in metres and seconds. Wings are maybe 10 cm thick, so then Stokes number = 109 d2 . This increases from 0.1 to 10 when droplets increase in size from 10 micrometres, to 100 micrometres. This means that most droplets 10 micrometres across or smaller will follow the air over the wing, while most 100 micrometre droplets or larger should go in more-or-less straight lines and splat into the front of the wing.

This can be quantified if we compute the width of the strip of air in front of the wing, where droplets of a particular diameter will hit the wing. I call this λ and it depends on the particle diameter through the Stokes number for that diameter. Langmuir and Blodgett used a 1940s differential analyser to compute many droplet trajectories and from this determined how λ varies with the Stokes number. Their result (from a 1946 US Army Air Forces Technical Report) is the blue curve below.

The red curve is from calculations I did yesterday evening using a Python code on my linux machine at home. The ratio λ/t goes from zero – so no particles collide – at small Stokes number, to close to one for large Stokes number. For droplets with Stokes numbers around 10 (which are droplets 100 micrometres in diameter), a wing of thickness t captures all the droplets of this size in the air over a strip of width about 85% of t.

Despite the impeccable CVs of the two scientists, before I compared the two sets of results I assumed with my vastly more powerful computer, that I could do better. But no, the two cuves are on top of each other.. I guess I should learn from this. Despite differential analysers being literally so old they are analog not digital, in the hands of good people you can excellent results with, by modern standards, ludicrously underpowered computational tools.