Bi-velocity hydrodynamics

Abstract

Theoretical evidence derived from linear irreversible thermodynamics (LIT) jointly with Burnett’s solution of Boltzmann’s gas-kinetic equation is used to show that fluid mechanics and transport processes in both gaseous and liquid continua require the use of two independent velocities rather than one in order to correctly quantify the physics of fluid motion. This finding, reflecting the coalescence of macroscopic and molecular perspectives, undermines the current foundations of continuum fluid mechanics. Of the two required context-specific velocities, one is the mass velocity v m appearing in the continuity equation. The other is the volume velocity v v entering into the constitutive equation P ⋅ v v for the mechanical rate-of-working term appearing in the energy equation, where it serves as the multiplier of the pressure tensor P . While the analysis involves only linear constitutive principles, the fundamental need for two independent velocities is noted to apply even in non-linear circumstances. A major consequence of these findings is that the Navier–Stokes–Fourier equations governing continuum fluid physics are incomplete for both single- and multi-component fluids. Our results are independently supported by the work of others based upon the use of conventional single-velocity arguments accompanied by ad hoc extensions of LIT. Our bi-velocity findings point to the existence of novel mechanodiffusive phenomena in fluid continua, entailing coupling between viscous flow and diffusion, whether referring to the diffusion of thermal energy in single-component non-isothermal fluids or of chemical species in inhomogeneous multicomponent mixtures.