A critical test of bivelocity hydrodynamics for mixtures

The present paper provides direct noncircumstantial evidence in support of the existence of a diffuse flux of volume $jv$ in mixtures. As such, it supersedes an earlier paper [H. Brenner, J. Chem. Phys. 132, 054106 (2010)], which offered only indirect circumstantial evidence in this regard. Given the relationship of the diffuse volume flux to the fluid’s volume velocity, this finding adds additional credibility to the theory of bivelocity hydrodynamics for both gaseous and liquid continua, wherein the term bivelocity refers to the independence of the fluid’s respective mass and volume velocities. Explicitly, the present work provides a new and unexpected linkage between a pair of diffuse fluxes entering into bivelocity mixture theory, fluxes that were previously regarded as constitutively independent, except possibly for their coupling arising as a consequence of Onsager reciprocity. In particular, for the case of a binary mixture undergoing an isobaric, isothermal, external force-free, molecular diffusion process we establish by purely macroscopic arguments—while subsequently confirming by purely molecular arguments—the validity of the ansatz $jv=(v¯1−v¯2)j1$ relating the diffuse volume flux $jv$ to the diffuse mass fluxes $j1(=−j2)$ of the two species and, jointly, their partial specific volumes $v¯1,v¯2$⁠. Confirmation of that relation is based upon the use of linear irreversible thermodynamic principles to embed this ansatz in a broader context, and to subsequently establish the accord thereof with Shchavaliev’s solution of the multicomponent Boltzmann equation for dilute gases [M. Sh. Shchavaliev, Fluid Dyn. 9, 96 (1974)]. Moreover, because the terms $v¯1$⁠, $v¯2$⁠, and $j1$ appearing on the right-hand side of the ansatz are all conventional continuum fluid-mechanical terms (with $j1$ given, for example, by Fick’s law for thermodynamically ideal solutions), parity requires that $jv$ appearing on the left-hand side of that relation also be a continuum term. Previously, diffuse volume fluxes, whether in mixtures or single-component fluids, were widely believed to be noncontinuum in nature, and hence of interest only to those primarily concerned with transport phenomena in rarefied gases. This demonstration of the continuum nature of bivelocity hydrodynamics suggests that the latter subject should be of general interest to all fluid mechanicians, even those with no special interest in mixtures.