Diffuse volume transport in fluids

The diffuse flux of volume jv in a single-component liquid or gas, the subject of this paper, is a purely molecular quantity defined as the difference between the flux of volume nv and the convective flux of volume nmvˆ carried by the flowing mass, with nm the mass flux, vˆ=1/ρ the specific volume, and ρ the mass density. Elementary statistical–mechanical arguments are used to derive the linear constitutive equation jv=DS∇lnρ, valid in near-equilibrium fluids from which body forces are absent. Here, DS is the fluid’s self-diffusion coefficient. The present derivation is based on Einstein’s mesoscopic Brownian motion arguments, albeit applied here to volume- rather than particle-transport phenomena. In contrast to these mesoscale arguments, all prior derivations were based upon macroscale linear irreversible thermodynamic (LIT) arguments. DS replaces the thermometric diffusivity α as the phenomenological coefficient appearing in earlier, ad hoc, derivations. The prior scheme based on α, which had been shown to accord with Burnett’s well-known gas-kinetic constitutive data for the heat flux and viscous stress, carries over intact to now show comparable accord of DS with these same data, since for gases the dimensionless Lewis number Le=α/DS is essentially unity. On the other hand for most liquids, where Le≫1, use of DS in place of α is shown to agree much better with existing experimental data for liquids. For the case of binary mixtures it is shown for the special case of isothermal, isobaric, force-free, Fick’s law-type molecular diffusion processes that jv=D∇lnρ, where D is the binary diffusion coefficient. In contrast with the preceding use in the single-component case of both mesoscopic and LIT models to obtain a constitutive equation for jv, the corresponding mixture result is derived here without use of any physical model whatsoever. Rather, the derivation effectively requires little more than the respective definitions of the diffuse volume- and Fickian mass-fluxes.