A GENERIC formalism for Korteweg-type fluids: II. Higher-order models and relation to microforces

In this paper, a second order model for Korteweg-type fluids is derived within the GENERIC formalism, whose standard application to Korteweg-type fluids is studied in detail in Part I of this series of papers. The higher-order model contains the second-order gradient of the density in the constitutive relation concerning internal energy, and the resulting Korteweg stress covers the original model proposed by Korteweg. This is in contrast to the model derived by Dunn and Serrin, which lacks the term of the second-order gradient of density. The higher-order model is thermodynamically consistent and includes interstitial working, as in Dunn and Serrin's model. The GENERIC formalism provides a fairly simple derivation of higher-order models for Korteweg-type fluids. Another possible formulation based on the concept of microforces introduced by Gurtin is also presented and discussed in relation to the GENERIC formalism.