On the development and generalizations of Cahn–Hilliard equations within a thermodynamic framework

Abstract

We provide a thermodynamic basis for the development of models that are usually referred to as “phase-field models” for compressible, incompressible, and quasi-incompressible fluids. Using the theory of mixtures as a starting point, we develop a framework within which we can derive “phase-field models” both for mixtures of two constituents and for mixtures of arbitrarily many fluids. In order to obtain the constitutive equations, we appeal to the requirement that among all admissible constitutive relations that which is appropriate maximizes the rate of entropy production (see Rajagopal and Srinivasa in Proc R Soc Lond A 460:631–651, 2004). The procedure has the advantage that the theory is based on prescribing the constitutive equations for only two scalars: the entropy and the entropy production. Unlike the assumption made in the case of the Navier–Stokes–Fourier fluids, we suppose that the entropy is not only a function of the internal energy and the density but also of gradients of the partial densities or the concentration gradients. The form for the rate of entropy production is the same as that for the Navier–Stokes–Fourier fluid. As observed earlier in Heida and Málek (Int J Eng Sci 48(11):1313–1324, 2010), it turns out that the dependence of the rate of entropy production on the thermodynamical fluxes is crucial. The resulting equations are of the Cahn–Hilliard–Navier–Stokes type and can be expressed both in terms of density gradients or concentration gradients. As particular cases, we will obtain the Cahn–Hilliard–Navier–Stokes system as well as the Korteweg equation. Compared to earlier approaches, our methodology has the advantage that it directly takes into account the rate of entropy production and can take into consideration any constitutive assumption for the internal energy (or entropy).