Incompressible flow

A flow is incompressible when the density of fluid parcels doesn't change measurably with pressure or motion. The continuity equation then reduces to ∇·v = 0 — velocity has zero divergence.

Liquids are almost perfectly incompressible at ordinary pressures (water's density changes by 0.5% under 100 atmospheres). Gases are incompressible in practice when the flow speed is well below the speed of sound (Mach number M ≲ 0.3). Above that, density variations become significant and full compressible flow theory is needed — shock waves, sonic booms, and all the hypersonic machinery.