Multiscale Problems in Three Applications - Abstract

Kraus, Christiane

Equilibrium conditions of liquid--vapour systems and the thermodynamical consistency of the van der Waals--Cahn--Hilliard phase model

We first revisit the classical problem of liquid--vapour systems of a single substance at constant temperature. We assume that the two--phase system is contained in a vessel with a conserved total mass of the substance, whereby the vessel can be conducted at constant volume or at constant pressure. Our emphasis will be on the influence of different boundary conditions on the resulting equilibria for the van der Waals--Cahn--Hilliard model and the corresponding sharp interface model.
In the second part of the talk we show the compatibility of the van der Waals--Cahn--Hilliard phase model to the local version of the the second law of thermodynamics, i.e. to the local entropy inequality. It is well known, that the entropy inequality allows for the appearance of $nabla rho$ in the free energy only, if either the energy flux or the entropy flux is represented by a non-classical form. The various contributions to the balance of the internal energy are in principle directly measurable, whereas the entropy flux is not accessible to direct measurements. For this reason we prefer to preserve the classical form of the energy balance. We identify a non-classical entropy flux which admits higher-order gradient contributions in the free energy and in the Korteweg stress tensor.