scholarly journals Existence of Weak Solutions for a Diffuse Interface Model for Viscous, Incompressible Fluids with General Densities

2009 ◽  
Vol 289 (1) ◽  
pp. 45-73 ◽  
Author(s):  
Helmut Abels
Keyword(s):  
Weak Solutions ◽  
Incompressible Fluids ◽  
Diffuse Interface ◽  
Interface Model ◽  
Diffuse Interface Model ◽  
Existence Of Weak Solutions

scholarly journals Global existence of weak solutions for a nonlocal model for two-phase flows of incompressible fluids with unmatched densities

2016 ◽  
Vol 26 (10) ◽  
pp. 1955-1993 ◽  
Author(s):  
Sergio Frigeri
Keyword(s):  
Weak Solutions ◽  
Type System ◽  
Diffuse Interface ◽  
Navier Stokes ◽  
Diffuse Interface Model ◽  
Two Phase ◽  
Existence Of Weak Solutions ◽  
Cahn Hilliard Equation ◽  
The One ◽  
Three Space

We consider a diffuse interface model for an incompressible isothermal mixture of two viscous Newtonian fluids with different densities in a bounded domain in two or three space dimensions. The model is the nonlocal version of the one recently derived by Abels, Garcke and Grün and consists in a Navier–Stokes type system coupled with a convective nonlocal Cahn–Hilliard equation. The density of the mixture depends on an order parameter. For this nonlocal system we prove existence of global dissipative weak solutions for the case of singular double-well potentials and non-degenerate mobilities. To this goal we devise an approach which is completely independent of the one employed by Abels, Depner and Garcke to establish existence of weak solutions for the local Abels et al. model.

A quasi-incompressible diffuse interface model with phase transition

2014 ◽  
Vol 24 (05) ◽  
pp. 827-861 ◽  
Author(s):  
Gonca L. Aki ◽  
Wolfgang Dreyer ◽  
Jan Giesselmann ◽  
Christiane Kraus
Keyword(s):  
Stokes Equations ◽  
Incompressible Fluids ◽  
The Other ◽  
Diffuse Interface ◽  
Navier Stokes ◽  
Interface Model ◽  
Diffuse Interface Model ◽  
Navier Stokes Equations ◽  
Interfacial Conditions ◽  
Two Component

This work introduces a new thermodynamically consistent diffuse model for two-component flows of incompressible fluids. For the introduced diffuse interface model, we investigate physically admissible sharp interface limits by matched asymptotic techniques. To this end, we consider two scaling regimes where in one case we recover the Euler equations and in the other case the Navier–Stokes equations in the bulk phases equipped with admissible interfacial conditions. For the Navier–Stokes regime, we further assume the densities of the fluids are close to each other in the sense of a small parameter which is related to the interfacial thickness of the diffuse model.

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