The Blesgen and Lowengrub-Truskinovsky Descriptions of Two-Phase Compressible Fluid Flow: Interstitial Working and a Reduction to Korteweg Theory

Abstract

Formulated by Blesgen and Lowengrub and Truskinovsky, the `Navier-Stokes-Allen-Cahn' (NSAC) equations and the `Navier-Stokes-Cahn-Hilliard' (NSCH) equations describe diphasic fluid flow, combining the conservation laws for mass, momentum, and energy with a balance law for the phases, which governs the concentration of one, and thus the other, phase as an order parameter. By contrast, the `Navier-Stokes-Korteweg' (NSK) theory for the dynamics of a one-phase capillary fluid, given by Dunn and Serrin following Korteweg and Slemrod, uses only the three said conservation laws and has the mass density itself as its order parameter. In a previous paper (ARMA, 2017), the authors considered the derivation of the NSAC, the NSCH, and the NSK equations, and showed that in the case that the two phase are incompressible with different specific volumes, both NSAC and NSCH reduce to versions of NSK. They realized only after that article was published that their assumptions tacitly corresponded to neglecting the so-called microforces. While the argumentation in that previous paper remains mathematically consistent, the microforces are a physical reality, and the purpose of the present note is to show that the reduction of NSAC and NSCH to NSK still holds when one does take them into account.