Phase-field descriptions of two-phase compressible fluid flow: Interstitial working and a reduction to Korteweg theory
First formulated by Truskinovsky [14] and Lowengrub and Truskinovsky [10], the ‘NavierStokes-Allen-Cahn’ equations1 and the ‘Navier-Stokes-Cahn-Hilliard’ equations describe diphasic fluid flow. They combine the conservation laws for mass, momentum, and energy with a balance law for the phases which governs the concentration of one (or the other) phase as an order parameter. By contrast, the ‘Navier-Stokes-Korteweg’ theory for the dynamics of a one-phase capillary fluid that Dunn and Serrin [3] gave following Korteweg [9] and Slemrod [12] uses only the three said conservation laws and has the mass density itself as its order parameter.2 In [6], the authors considered the derivation of the NSAC, the NSCH, and the NSK equations, and showed that in the case that the two phases are incompressible with different specific volumes, both NSAC and NSCH reduce to versions of NSK. However, as we realized only after that article was published, our assumptions in [6] tacitly correspond to neglecting the so-called microforces [8]. While the argumentation in [6] seems 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. Also in NSAC and NSCH, the microforces do what Dunn and Serrin named interstitial work. In Section 2 we study the role that this particular effect plays in the entropy production along particle paths. Section 3 serves to prove the reduction property of NSAC and NSCH with interstitial work. Section 4 discusses the objectivity of the three models briefly. In the sequel, we use notation from [6] and the results of Section 1 of that paper without defining / stating them again. Since the publication of [6] we also noticed that already Korteweg himself realized his tensor should be useful in connection with incompressible phases — the modern NSAC and NSCH equations would be the “autres d’une complication plus grande encore”3 Korteweg envisaged as generalizations of what is nowadays called ‘dynamic Korteweg theory’.