Non-dimensionalization of the Compressible Navier-Stokes Equation by Pressure Wavelength and Period revealing its Singularity

Fluid particles oscillating in compressible fluid field will produce a pressure (density) wave. The wave propagates in the field by a finite speed – the wave speed of c. Any function where the x and t dependence is of the form (kx - \(\omega\)t) (or of the form f(x-ct) and g(x+ct)) represents a traveling wave of some shape. The x and t are no longer independent variables, rather, they are inter-dependent. They are related by the wave propagating speed of c through a dimensionless scalar function ̶ wave phase function of \(\phi(x,t) = kx \pm \omega t,\) where space and time are nondimensionalized by wavelength (wave number) and period (frequency). All the inertial observers perceive the same dimensionless wave phase function, \(\phi(x,t)\), at a given point in space and time if they have relative motions. If the wavelength and period are picked out as a measuring unit to quantify the length and time interval, the physical values of the length and time interval must be correspondingly enlarged or shortened for different initial frames, to ensure the wave phase function to be an invariant dimensionless scalar function. The classical compressible Navier-stokes equation contains pressure and density terms, if it is non-dimensionalized by the pressure wavelength and period in an inertial frame; the resulting equation contains a Reynolds number, where the wavelength is its scaling length. Its value will change following the fluid particle motion, relative to rest frame (Lab frame); The wavelength will be shorter (frequency will be higher) if the fluid particle flows toward an observer. The flow will blow up when the flow velocity approaches wave propagation velocity. Furthermore, it shows mass-energy equivalence can be expressed as \(pV = mc^{2}\) in the co-moving reference frame (rest frame).