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We develop a decoupled, first-order, fully discrete, energy-stable scheme for the Cahn-Hilliard-Navier-Stokes equations. This scheme calculates the Cahn-Hilliard and Navier-Stokes equations separately, thus effectively decoupling the entire…

Numerical Analysis · Mathematics 2025-06-25 Haijun Gao , Xi Li , Minfu Feng

The dynamics of phase separation for a binary fluid subjected to a uniform shear are solved exactly for a model in which the order parameter is generalized to an n-component vector and the large-n limit taken. Characteristic length scales…

Statistical Mechanics · Physics 2009-10-31 N. P. Rapapa , A. J. Bray

We study a diffusion model of phase field type, consisting of a system of two partial differential equations encoding the balances of microforces and microenergy; the two unknowns are the order parameter and the chemical potential. By a…

Analysis of PDEs · Mathematics 2011-03-24 Pierluigi Colli , Gianni Gilardi , Paolo Podio-Guidugli , Juergen Sprekels

A computationally efficient, low order finite element formulation is developed for modelling the Navier-Stokes-Cahn-Hilliard equations, which have been established as a promising phase field modelling approach for simulation of immiscible…

Computational Physics · Physics 2019-11-18 Aleksander Lovrić , Wulf G. Dettmer , Djordje Perić

We study a Cahn--Hilliard two-phase model describing the flow of two viscoelastoplastic fluids, which arises in geodynamics. A phase-field variable indicates the proportional distribution of the two fluids in the mixture. The motion of the…

Analysis of PDEs · Mathematics 2025-10-01 Fan Cheng , Robert Lasarzik , Marita Thomas

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

The paper considers a thermodynamically consistent phase-field model of a two-phase flow of incompressible viscous fluids. The model allows for a non-linear dependence of fluid density on the phase-field order parameter. Driven by…

Numerical Analysis · Mathematics 2023-09-27 Yerbol Palzhanov , Alexander Zhiliakov , Annalisa Quaini , Maxim Olshanskii

We extend the two-scale expansion approach of periodic homogenization to include time scales and thus can tackle the full instationary Navier-Stokes-Cahn-Hilliard model at the pore scale as microscale. Time scale separation allows us to…

Fluid Dynamics · Physics 2021-03-04 Stefan Metzger , Peter Knabner

We consider a diffuse interface model for incompressible isothermal mixtures of two immiscible fluids with matched constant densities. This model consists of the Navier-Stokes system coupled with a convective nonlocal Cahn-Hilliard equation…

Analysis of PDEs · Mathematics 2014-04-16 Sergio Frigeri , Maurizio Grasselli , Elisabetta Rocca

Fluid mixture models are essential for describing a wide range of physical phenomena, including wave dynamics and spinodal decomposition. However, there is a lack of consensus in the modeling of compressible mixtures, with limited…

Fluid Dynamics · Physics 2025-04-01 M. F. P. ten Eikelder , E. H. van Brummelen , D. Schillinger

In this paper, we present an efficient numerical algorithm for solving the time-dependent Cahn--Hilliard--Navier--Stokes equations that model the flow of two phases with different densities. The pressure-correction step in the projection…

Numerical Analysis · Mathematics 2020-11-02 Chen Liu , Deep Ray , Christopher Thiele , Lu Lin , Beatrice Riviere

In this paper, we introduce a model describing the dynamic of vesicle membranes within an incompressible viscous fluid in $3D$ domains. The system consists of the Navier-Stokes equations, with an extra stress tensor depending on the…

Analysis of PDEs · Mathematics 2017-10-10 Blanca Climent-Ezquerra , Francisco Guillén-González

A new theory on gas-liquid phase transition is given. The new idea is that the total intermolecular potential energy for a classical system in equilibrium is relative with the average distance of molecules. A new space homogeneity…

Statistical Mechanics · Physics 2007-05-23 Yuan-Xing Gui

We consider a differential model describing nonisothermal fast phase separation processes taking place in a three-dimensional bounded domain. This model consists of a viscous Cahn-Hilliard equation characterized by the presence of an…

Analysis of PDEs · Mathematics 2007-05-23 Maurizio Grasselli , Hana Petzeltova , Giulio Schimperna

The two-phase Navier-Stokes Cahn-Hilliard (NSCH) mixture model is a key framework for simulating multiphase flows with non-matching densities. Developing fully discrete, energy-stable schemes for this model remains challenging, due to the…

Numerical Analysis · Mathematics 2026-02-05 Aaron Brunk , Marco F. P. ten Eikelder

We show well-posedness of a diffuse interface model for a two-phase flow of two viscous incompressible fluids with different densities locally in time. The model leads to an inhomogeneous Navier-Stokes/Cahn-Hilliard system with a solenoidal…

Analysis of PDEs · Mathematics 2020-10-14 Helmut Abels , Josef Weber

We analyze a Navier-Stokes-Cahn-Hilliard model for viscous incompressible two-phase flows where the mechanisms of chemotaxis, active transport and reaction are taken into account. The evolution system couples the Navier-Stokes equations for…

Analysis of PDEs · Mathematics 2024-06-03 Jingning He , Hao Wu

We consider the numerical approximations of a two-phase hydrodynamics coupled phase-field model that incorporates the variable densities, viscosities and moving contact line boundary conditions. The model is a nonlinear, coupled system that…

Numerical Analysis · Mathematics 2017-02-15 Haijun Yu , Xiaofeng Yang

The phase separation of an isothermal incompressible binary fluid in a porous medium can be described by the so-called Brinkman equation coupled with a convective Cahn-Hilliard (CH) equation. The former governs the average fluid velocity…

Analysis of PDEs · Mathematics 2016-01-14 Francesco Della Porta , Maurizio Grasselli

Hamiltonian particle systems may exhibit non-linear hydrodynamic phenomena as the time evolution of the density fields of energy, momentum, and mass. In this Letter, an exact equation describing the time evolution is derived assuming the…

Statistical Mechanics · Physics 2014-03-18 Shin-ichi Sasa