Alessia Berti

A mathematical model for phase separation: a generalized Cahn-Hilliard equation

In this paper we present a mathematical model to describe the phenomenon of phase separation, which is modelled as space regions where an order parameter changes smoothly. The model proposed, including thermal and mixing effects, is deduced for an incompressible fluid, so the resulting differential system couples a generalized Cahn–Hilliard equation with the Navier–Stokes equation. Its consistency with the second law of thermodynamics in the classical Clausius–Duhem form is finally proved. Copyright

A Phase-Field Model for Liquid–Vapor Transitions

Abstract Starting from the mesoscopic description of the state equations for the vapor and liquid pure phases of a single chemical species, we propose a phase-field model ruling the liquid–vapor phase transition. Two different phases are separated by a thin layer, rather than a sharp interface, where the phase field changes abruptly from 0 to 1. All thermodynamic quantities are allowed to vary inside the transition layer, including the mass density. The approach is based on an extra entropy flux which is proved to be non-vanishing inside the transition layer only. Unlike classical phase-field models, the kinetic equation for the phase variable is obtained as a consequence of thermodynamic restrictions and it depends only on the rescaled free enthalpy. The system turns out to be thermodynamically consistent and accounts for both temperature and pressure variations during the evaporation process. Few commonly accepted assumptions allow us to obtain the explicit expression of the Gibbs free enthalpy and the Clausius–Clapeyron formula. As a consequence, the customary form of the vapor pressure curve is recovered.

A thermodynamically consistent Ginzburg–Landau model for superfluid transition in liquid helium

In this paper, we propose a thermodynamically consistent model for superfluid-normal phase transition in liquid helium, accounting for variations of temperature and density. The phase transition is described by means of an order parameter, according to the Ginzburg–Landau theory, emphasizing the analogies between superfluidity and superconductivity. The normal component of the velocity is assumed to be compressible, and the usual phase diagram of liquid helium is recovered. Moreover, the continuity equation leads to a dependence between density and temperature in agreement with the experimental data.

Global and exponential attractors for a Ginzburg-Landau model of superfluidity

The long-time behavior of the solutions for a non-isothermal model in superfluidity is investigated. The model describes the transition between the normal and the superfluid phase in liquid 4 He by means of a non-linear differential system, where the concentration of the superfluid phase satisfies a non-isothermal Ginzburg-Landau equation. This system, which turns out to be consistent with thermodynamical principles and whose well-posedness has been recently proved, has been shown to admit a Lyapunov functional. This allows to prove existence of the global attractor which consists of the unstable manifold of the stationary solutions. Finally, by exploiting recent techinques of semigroups theory, we prove the existence of an exponential attractor of finite fractal dimension which contains the global attractor.

Existence and uniqueness for the reflection and transmission problem in stratified electromagnetic media

The reflection-transmission problem of time-harmonic waves in a stratified electromagnetic medium is investigated. The waves are sent from upward or downward with oblique incidence. By means of the energy flux, up-going and down-going waves are distinguished and the reflection and transmission matrices are introduced. When the solid occupies a strip between two homogeneous media, the existence and uniqueness of the reflected and transmitted waves are proved. The same conclusions are obtained for a dielectric without memory extended in the whole space.

A Ginzburg–Landau model for material aging depending on temperature

ABSTRACT We consider a model describing the behavior of a body subject to aging and fatigue. These phenomena are supposed to be affected by both mechanical and thermal effects. The material is assumed to be viscoelastic where the stress–strain relation is based on a new fractional derivative proposed in Caputo and Fabrizio. The order of derivative is regarded as a new variable whose evolution is ruled by a Ginzburg–Landau equation. The model also includes an evolutive equation for the temperature deducing from the first law of thermodynamics. In this article, thermodynamic compatibility is shown and some numerical simulations are performed.

Transition and separation process in brine channels formation

In this paper, we discuss the formation of brine channels in sea ice. The model includes a time-dependent Ginzburg-Landau equation for the solid-liquid phase change, a diffusion equation of the Cahn-Hilliard kind for the solute dynamics, and the heat equation for the temperature change. The macroscopic motion of the fluid is also considered, so the resulting differential system couples with the Navier-Stokes equation. The compatibility of this system with the thermodynamic laws and a maximum theorem is proved.