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>The incompressible Euler equations model a fluid as a two-valued field. This means that at every point in space, the field has two values, density and velocity

I don't get it. If the fluid is incompressible, how can density have a value at every point in space? Isn't it just a constant?

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The density can be constant, but it doesn't have to be. If the density field starts out with some variation in it, then those variations move around as the fluid flows. Incompressibility just means that those density variations can't get bigger or smaller, they can only move, shear, and rotate.

When you work with near-supercritical and supercritical fluids under laboratory conditions, you can turn the pump by hand and you feel when the density hits the ceiling.

So you know something is up.

Systems would be modeled mathematically using a fluid's individual component values, but we were paid for the real-world laboratory data.

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