We present a hybrid level–set–Cahn–Hilliard model for the simulation of two phase flow in micro structures. Our method combines accurate contact line dynamics in the Cahn–Hilliard equation with the computationally cheaper level set method in regions where not the full Cahn–Hilliard dynamics are necessary to describe the flow. This is realized by employing the Cahn–Hilliard model close to rigid ...

We consider a diffusive interface surface tension model under compressible flow. The equation of interest is the Cahn-Hilliard or Allen-Cahn equation with advection by a non-divergence free velocity field. These are two reduced models which show important properties of the full-scale surface tension model. We prove that both model problems are well-posed. We are especially interested in the beh...

In this paper we develop local discontinuous Galerkin (LDG) methods for the fourth-order nonlinear Cahn-Hilliard equation and system. The energy stability of the LDG methods is proved for the general nonlinear case. Numerical examples for the Cahn-Hilliard equation and the Cahn-Hilliard system in one and two dimensions are presented and the numerical results illustrate the accuracy and capabili...

We derive a Fractional Cahn-Hilliard Equation (FCHE) by considering a gradient flow in the negative order Sobolev space H−α, α ∈ [0, 1] where the choice α = 1 corresponds to the classical Cahn-Hilliard equation whilst the choice α = 0 recovers the Allen-Cahn equation. The existence of a unique solution is established and it is shown that the equation preserves mass for all positive values of fr...

In this paper, we apply the spectral method to approximate the solutions of Cahn-Hilliard equation, which is a typical class of nonlinear fourth-order diffusion equations. Diffusion phenomena is widespread in the nature. Therefore, the study of the diffusion equation caught wide concern. Cahn-Hilliard equation was proposed by Cahn and Hilliard in 1958 as a mathematical model describing the diff...

The process of phase separation of binary systems is described by the Cahn-Hilliard equation. The main objective of this article is to give a classification on the dynamic phase transitions for binary systems using either the classical Cahn-Hilliard equation or the Cahn-Hilliard equation coupled with entropy, leading to some interesting physical predictions. The analysis is based on dynamic tra...

The grounding process relies on the evidence that speakers give about their understanding (Clark and Schaefer, 1989). However in existing models of grounding (Cahn, 1992; Cahn and Brennan, 1999; Traum, 1999) evidence of understanding is assumed to be symmetrically and synchronously shared by the speakers. We propose a grounding model based on (Cahn, 1992) that removes these simplifications; we ...

We carry out rigorous error analysis for some energy stable time discretization schemes developed in [20] for a Cahn-Hilliard phase-field model and in [19] for an Allen-Cahn phase-field model.

We propose a novel second order in time numerical scheme for Cahn-Hilliard-NavierStokes phase field model with matched density. The scheme is based on second order convex-splitting for the Cahn-Hilliard equation and pressure-projection for the Navier-Stokes equation. We show that the scheme is mass-conservative, satisfies a modified energy law and is therefore unconditionally stable. Moreover, ...

In this article, we study the initial-boundary-value problem for a sixth-order Cahn-Hilliard type equation ut = D μ, μ = γDu− a(u)Du− a′(u) 2 |Du| + f(u) + kut, which describes the separation properties of oil-water mixtures, when a substance enforcing the mixing of the phases is added. The optimal control of the sixth order Cahn-Hilliard type equation under boundary condition is given and the ...