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

TL;DR

This paper develops a comprehensive mathematical model combining a generalized Cahn-Hilliard equation with Navier-Stokes to describe phase separation in incompressible fluids, incorporating thermal and mixing effects while ensuring thermodynamic consistency.

Contribution

It introduces a novel coupled model that integrates phase separation dynamics with fluid flow and thermodynamics, extending previous models with a generalized Cahn-Hilliard formulation.

Findings

Model is consistent with the second law of thermodynamics.

Couples phase separation with fluid dynamics in a unified framework.

Provides a basis for further analytical and numerical studies.

Abstract

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.