Abstract Dimensional analysis based on the principle of dimensional invariance allows the introduction of dimensionless numbers, like the famous Reynolds number, via the application of the Vaschy–Buckingham theorem. Dynamic similarity and self-similarity are described. The analysis of the dimensionless compressible Navier–Stokes equations shows how the incompressible equations are recovered when the Mach number goes to zero. The nature of pressure in the compressible and incompressible cases is broadly discussed.