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

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.

On a Chebyshev accelerated splitting iteration method with application to two‐by‐two block linear systems

“…In other cases, it is not easy to find the infimum of the condition number. Instead, we offer the possibly pessimistic bound in (18).…”

“…16 We refer to the study of Axelsson et al 16 for a more detailed introduction and comparison of such effective preconditioners for solving the PDE-constrained optimization problems with Possion control and convection-diffusion control. The linear system (1) also arises in finite-element discretization and first-order linearization of the two-phase flow problems based on Cahn-Hilliard equation [17][18][19] and so on.…”

Generalized transport model for phase transition with memory

“…(4) by taking ε > 0, κ > 0 and τ = 0: ∂u ∂t = ∇ 2 Φ(u) − κ∇ 4 u + ε∇ 2 ∂u ∂t (8) which is known as the viscous Cahn-Hilliard equation [32] and its consistency with the second law of thermodynamics has been proved [33]. We notice that in literature, a numerical solution to the viscous Cahn-Hilliard equation is seldom given and how the various effects interplay in a dynamic phase transition process is not investigated.…”

Phase separation in quasi-incompressible fluids: Cahn–Hilliard model in the Cattaneo–Maxwell framework

“…Typically it takes place when the mixture is quickly cooled below a critical value of the temperature where the mixture can no longer exist in equilibrium in its homogeneous state ( [10]). Even the velocity can influence the miscibility properties of the mixture (see [2,3,10,15]).…”