The Navier-Stokes-Fourier equations revisited

Continuum mechanics principles are applied to standard balance equations assuming constitutive equations carried out to first derivatives of density, temperature, and velocity. Using the principles of isotropy, frame invariance, and equipresence, constitutive equations are ontained for the Helmholtz free energy, entropy, entropy flux, extra entropy flux, and Cauchy stress tensor that satisfy the Clausius-Duhem inequality. The analysis provides the most general linear constitutive equations of a fluid and leads to a new definition of the bulk viscosity. The equation for the Cauchy stress tensor contains a term that is missing in the well-known Navier-Stokes-Fourier equations. The additional term, when applied to an ideal gas, appears to be due to the local time relaxation of molecular vibrational energy, and when applied to a liquid, appears to be due to the relaxation of potential energy between molecules. Additional terms in the constitutive equation of entropy are also obtained that are due to the relative time rate of change of local volume.