Temperature dependent extensions of the Cahn-Hilliard equation

Abstract

The Cahn-Hilliard equation is a fundamental model that describes phase separation processes of binary mixtures. In this paper we focus on the dynamics of these binary media, when the underlying temperature is not constant. The aim of this paper is twofold. We first derive two distinct models that extend the classical Cahn-Hilliard equation with an evolutionary equation for the absolute temperature. Secondly, we analyse the local well-posedness of classical solution for one of these systems. Our modelling introduces the systems of PDEs by means of a general and unified formalism. This formalism couples standard principles of mechanics together with the main laws of thermodynamics. Our work highlights how certain assumptions on the transport of the temperature effect the overall physics of the systems. The variety of these thermodynamically consistent models opens the question of which one should be more appropriate. Our analysis shows that one of the derived models might be more desirable to the well-posedness theory of classical solutions, a property that might be natural as a selection criteria. We conclude our paper with an overview and comparison of our modelling formalism with some equations, which were previously derived in literature.

Global well-posedness of a Navier–Stokes–Cahn–Hilliard system with chemotaxis and singular potential in 2D

“…It is a reasonable approximation to the real temperature-dependent model, where the chemical potential should be dependent of the temperature. A more complicated model with temperature-dependent singular potential has been obtained in a recent literature [5].…”

Some Recent Advances in Energetic Variational Approaches

“…Following the same procedure, one can show that the entropy production is same as in (95). We refer the interested reader to [98] for a detailed derivation.…”

On a Cahn-Hilliard system with source term and thermal memory

“…Next, employing micro-force balance theory, Miranville and Schimperna proposed a further derivation in [29], and the well-posedness of a related system has been addressed in [26]. Moreover, we point out the recent contribution [15] by De Anna et al, where two new thermodynamically consistent models related to nonisothermal Cahn-Hilliard systems have been derived. Finally, we refer to [18,19] for some mathematical results on a relaxed version of the above systems endowed with dynamic boundary conditions.…”