Ideal flow
The theory that solves everything
Throw away viscosity and assume nothing is spinning, and fluid mechanics collapses into a linear problem with closed-form answers. The price is one term, and the term turns out to matter more than everything kept.
The exact theory says nothing has any drag
Solve the flow past a body in a fluid with no viscosity and the answer is beautiful, closed-form, and predicts that a cyclist needs no legs and an airliner no engines. This is not a small error, and it is the most useful failure in the subject.
Flows add up
The equations of ideal flow are linear, so solutions can be laid on top of one another. A uniform stream plus a doublet produces a cylinder that nobody put there, and almost every classical result is built this way.
Fast means low pressure
The trade between speed and pressure is the most useful relation in the subject and the most misused. Where it comes from, what it costs, and why the pressure over a wing is negative almost everywhere.