Kinematics of volume transport

The transport and production of the generally non-conserved extensive property of volume V within a flowing gas or liquid is addressed. Specifically, a convective/diffusive/production-type transport equation, ρDmv^/Dt+∇·jv=πv, governing the transport of volume, is derived for the evolution of the fluid's specific volume v^=1/ρ. Here, ρ is the density, Dm/Dt the material derivative, jv the diffusive flux density vector of volume, and πv the temporal rate of production of volume. This equation governs the transport of volume in precisely the same sense as do its counterparts governing the transport of other extensive properties, including mass, individual species mass in multicomponent mixtures, energy, entropy, momentum, and the like. Constitutive expressions are developed for both jv and πv in terms of the local specific-volume gradient ∇v^ and the physicochemical properties of the extensive property being transported, which volume inseparably accompanies. The resulting expressions are valid for situations wherein the state of the system is governed by a single independent variable, such as composition in an isothermal binary mixture, or temperature in a single-component system (the fluid in both cases being assumed effectively isobaric, consistent with the local mechanical equilibrium hypothesis of irreversible thermodynamics). The phenomenological volume diffusivity coefficient αv appearing in the constitutive equation jv=-αvρ∇v^ for the diffusive flux density is shown to be equal to either the binary molecular diffusivity D in the former case or to the thermometric diffusivity α in the latter case. In the important dual circumstances where a “law of additive volumes” is applicable to the fluid, and where the extensive property accompanying the volume being transported is a conserved property, volume itself becomes a conserved property. In that case one has that πv=0 for the production term in the volume transport equation. Inseparably related to the notion of a diffusive flux density jv of volume is the concept of a convective volume velocity vv. The latter velocity is shown to be related via the formula vv=vm+jv to the fluid's ubiquitous barycentric velocity vm appearing in the continuity equation governing the transport of mass. In the binary mixture case these two velocities are, respectively, identified with the well-known volume- and mass-average velocities of the fluid.