Beyond Navier–Stokes

The Navier–Stokes–Fourier (NSF) equations are shown to be strictly applicable only to incompressible flows, namely those involving fluids whose density is uniform throughout. Linear irreversible thermodynamic principles are used to derive an amended set of mass, momentum, and energy equations applicable to all fluids, compressible and incompressible, liquids as well as gases. These reduce to those of NSF in the incompressible limit. A modification of Fourier’s heat-conduction constitutive law is also required as a consequence of the mass/volume velocity difference, wherein the heat flux is now defined in relation to the Second- rather than First-law of thermodynamics; that is, the heat flux is expressed in terms of entropy transport rather than energy transport, with the distinction between the two fluxes vanishing for incompressible flows. When solved subject to a no-slip boundary condition imposed on the fluid’s volume velocity rather than on its mass velocity, the amended NSF equations are noted to furnish results in accord with a variety of experimental data for isothermal and nonisothermal, compressible and incompressible, gas and liquid flows. The difference between the fluid’s volume and mass velocities, namely the diffuse flux of volume, is shown to constitute the hydrodynamic-level manifestation of the fluid’s biased Brownian motion, with the bias arising from the inhomogeneity in mass distribution arising in compressible fluids from temperature or pressure gradients. Previously, continuum hydrodynamics, as embodied in the Navier–Stokes–Fourier equations, has overlooked the fundamental contribution to fluid mechanics emanating from biased Brownian motion.