A volume-based description of gas flows with localised mass-density   variations
  We reconsider some fundamental aspects of the fluid mechanics model, and the derivation of continuum flow equations from gas kinetic theory. Two topologies for fluid representation are presented, and a set of macroscopic equations are derived through a modified version of the classical Boltzmann kinetic equation for monatomic gases. The free volumes around the gaseous molecules are introduced into the set of kinetic microscopic parameters. Our new description comprises four, rather than three, conservation equations; the classical continuity equation, which conflates actual mass-density and number-density in a single equation, has been split into a conservation equation of mass (which involves only the classical number-density of the gaseous particles) and an evolution equation purely of the mass-density (mass divided by the actual volume of the fluid). We propose this model as a better description of gas flows displaying non-local-thermodynamic-equilibrium (rarefied flows), flows with relatively large variations of macroscopic properties, and/or highly compressible fluids/flows. 
S. Kokou Dadzie, Jason M. Reese and Colin R. McInnes