Extensions to the Navier–Stokes equations

Historically, the mass conservation and the classical Navier-Stokes equations were derived in co-moving reference frame. It is shown that the mass conservation and Navier-Stokes equations are Galilean invariant - they are valid in any arbitrary inertial reference frame. From the mass conservation and Navier-Stokes equations, we can derive a wave equation, which contains the speed of pressure wave as its parameter. This parameter is independent of the speed of the source - the fluid velocity. The speed of pressure wave is reference frame independent. It is well known that Lorentz transformation ensures wave speed invariant in all inertial frames, and Lorentz invariance holds for different inertial observers. Based on these arguments, the Navier-Stokes equations (conservation law for the energy-momentum) can be written in any inertial reference frame, they are transformed from one reference frame into another with the help of the Lorentz transformation. The key issue is that the Lorentz factor is parametrized by the local Mach number. In the instantaneous co-moving reference frame, these equations will degrade to the classical Navier-Stokes equations - the limit of the non-relativistic ones. For subsonic flow, the classical Navier-Stokes equations are only valid as an approximation at very low Mach number. For supersonic flow, namely when the Mach number is greater than one, the Lorentz factor becomes an imaginary number, a faster-than-wave-speed reference frame (complex valued Lorentz transformation) should be applied to solve the supersonic flow problems. The general Navier-Stokes equations embody an intrinsic singularity when local Mach number is equal to one.