A constitutive model for volume-based descriptions of gas flows
  We derive the pressure tensor and the heat flux to accompany the new macroscopic conservation equations that we developed previously in a volume-based kinetic framework for gas flows. This kinetic description allows for expansion or compression of a fluid element, and is characterized by a flux of volume that conventional modelling does not account for. A new convective form of transport arises from this new approach, which is due purely to macroscopic expansion or compression. It supplements the conventional transport processes, which are transport due purely to mass motion (classical convective transport) and transport due purely to random motion (diffusive transport). In addition, we show that the thermodynamic parameters of the fluid (temperature and pressure) appear with new non-equilibrium contributions that depend on density variations in the gas. Our new model may be useful for describing gas flows that display non-local-thermodynamic-equilibrium (rarefied gas flows), flows with relatively large variations of macroscopic properties, and/or highly compressible fluids and flows. 
S. Kokou Dadzie and Jason M. Reese