Korteweg-Type Fluids and Thermodynamic Modelling via Higher-Order Gradients

This paper investigates the modelling of Korteweg-type fluids and hence the dependence of the stress tensor on gradients of mass density. This topic, originating from the need for describing capillarity effects, is mainly of interest in connection with nanosystems where the mean free path may be comparable with the geometric dimensions of the system. In addition to the Korteweg fluid model, the paper gives a review of the stress tensor function arising in quantum fluid hydrodynamics. Next, thermodynamic consistency is established for a fluid involving first- and second-order density gradients. The modelling investigated is a generalization of the classical Korteweg fluid and allows a better understanding of previous thermodynamic restrictions. The restrictions determined for the general scheme with second-order gradients are applied to the particular cases of the Korteweg fluid and the quantum fluid. Further, to allow for discontinuity wave solutions with finite speed of propagation, a model is established which involves higher-order derivatives and reduces to the Korteweg fluid in stationary conditions.