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The Cahn--Hilliard equation is one of the most common models to describe phase separation processes of a mixture of two materials. For a better description of short-range interactions between the material and the boundary, various dynamic…

Analysis of PDEs · Mathematics 2021-04-27 Patrik Knopf , Kei Fong Lam , Chun Liu , Stefan Metzger

In this paper, we study the longtime asymptotic behavior of a phase separation process occurring in a three-dimensional domain containing a fluid flow of given velocity. This process is modeled by a viscous convective Cahn-Hilliard system,…

Analysis of PDEs · Mathematics 2018-03-13 Pierluigi Colli , Gianni Gilardi , Jürgen Sprekels

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…

Mathematical Physics · Physics 2011-02-08 Alessia Berti , Ivana Bochicchio

The Cahn-Hilliard equation is the most common model to describe phase separation processes of a mixture of two components. For a better description of short-range interactions of the material with the solid wall, various dynamic boundary…

Analysis of PDEs · Mathematics 2020-10-20 Harald Garcke , Patrik Knopf

We use the Navier-Stokes-Cahn-Hilliard model equations to simulate phase separation with flow. We study coarsening - the growth of extended domains wherein the binary mixture phase separates into its component parts. The coarsening is…

Fluid Dynamics · Physics 2018-10-17 Aurore Naso , Lennon O'Naraigh

The Cahn-Hilliard equation is a fundamental model that describes phase separation processes of two-phase flows or binary mixtures. In recent years, the dynamic boundary conditions for the Cahn-Hilliard equation have been proposed and…

Dynamical Systems · Mathematics 2024-12-12 Shuting Gu , Ming Xiao , Rui Chen

In this paper we devise and analyze a mixed finite element method for a modified Cahn-Hilliard equation coupled with a non-steady Darcy-Stokes flow that models phase separation and coupled fluid flow in immiscible binary fluids and diblock…

Numerical Analysis · Mathematics 2013-12-24 Amanda E. Diegel , Xiaobing H. Feng , Steven M. Wise

The Cahn--Hilliard equation is a fundamental model that describes phase separation processes of binary mixtures. In recent years, several types of dynamic boundary conditions have been proposed in order to account for possible short-range…

Analysis of PDEs · Mathematics 2023-07-28 Chun Liu , Hao Wu

In this paper we propose a mathematical model of phase separation for a quasi-incompressible binary mixture where the spinodal decomposition is induced by an heat flux governed by the Cattaneo-Maxwell equation. As usual, the phase…

Mathematical Physics · Physics 2013-06-04 A. Berti , I. Bochicchio , M. Fabrizio

We introduce a new measure of coarseness for characterizing phase separation processes such as those described by Cahn--Hilliard equations. An advantage of our measure is that it remains consistent throughout the evolution, including for…

Analysis of PDEs · Mathematics 2026-01-22 Peter Howard , Adam Larios , Quyuan Lin

The Cahn--Hilliard equation is one of the most common models to describe phase segregation processes in binary mixtures. In recent times, various dynamic boundary conditions have been introduced to model interactions of the materials with…

Analysis of PDEs · Mathematics 2021-10-12 Patrik Knopf , Andrea Signori

An upper bound on the mixing efficiency is derived for a passive scalar under the influence of advection and diffusion with a body source. For a given stirring velocity field, the mixing efficiency is measured in terms of an equivalent…

Fluid Dynamics · Physics 2007-12-12 Jean-Luc Thiffeault , Charles R. Doering , John D. Gibbon

The mixing efficiency of a flow advecting a passive scalar sustained by steady sources and sinks is naturally defined in terms of the suppression of bulk scalar variance in the presence of stirring, relative to the variance in the absence…

Fluid Dynamics · Physics 2011-05-03 Tiffany A. Shaw , Jean-Luc Thiffeault , Charles R. Doering

A proof of convergence is given for bulk--surface finite element semi-discretisation of the Cahn--Hilliard equation with Cahn--Hilliard-type dynamic boundary conditions in a smooth domain. The semi-discretisation is studied in the weak…

Numerical Analysis · Mathematics 2020-12-01 Paula Harder , Balázs Kovács

In this paper we consider the Cahn-Hilliard equation endowed with Wentzell boundary condition which is a model of phase separation in a binary mixture contained in a bounded domain with permeable wall. Under the assumption that the…

Analysis of PDEs · Mathematics 2008-04-14 Hao Wu

In this paper we deal with a doubly nonlinear Cahn-Hilliard system, where both an internal constraint on the time derivative of the concentration and a potential for the concentration are introduced. The definition of the chemical potential…

Analysis of PDEs · Mathematics 2020-12-11 Elena Bonetti , Pierluigi Colli , Luca Scarpa , Giuseppe Tomassetti

In this paper, we consider the numerical approximations for the commonly used binary fluid-surfactant phase field model that consists two nonlinearly coupled Cahn-Hilliard equations. The main challenge in solving the system numerically is…

Numerical Analysis · Mathematics 2017-03-20 Xiaofeng Yang

In this paper, we investigate a distributed optimal control problem for a convective viscous Cahn-Hilliard system with dynamic boundary conditions. Such systems govern phase separation processes between two phases taking place in an…

Analysis of PDEs · Mathematics 2017-09-08 Pierluigi Colli , Gianni Gilardi , Jürgen Sprekels

A Cahn-Hilliard-type theory for hydrodynamic fluctuations is proposed that gives a quantitative description of the slowly evolving spatial correlations and structures in density and flow fields in the early stages of evolution of freely…

Statistical Mechanics · Physics 2009-10-31 T. P. C. van Noije , M. H. Ernst

The mixing of binary fluids by stirrers is a commonplace procedure in many industrial and natural settings, and mixing efficiency directly translates into more homogeneous final products, more enriched compounds, and often substantial…

Fluid Dynamics · Physics 2022-08-12 Maximilian F. Eggl , Peter J. Schmid