On an incompressible Navier–Stokes/Cahn–Hilliard system with degenerate mobility

Abstract

We prove existence of weak solutions for a diffuse interface model for the flow of two viscous incompressible Newtonian fluids in a bounded domain by allowing for a degenerate mobility. The model has been developed by Abels, Garcke and Grün for fluids with different densities and leads to a solenoidal velocity field. It is given by a nonhomogeneous Navier-Stokes system with a modified convective term coupled to a Cahn-Hilliard system, such that an energy estimate is fulfilled which follows from the fact that the model is thermodynamically consistent.

On Convergent Schemes for Diffuse Interface Models for Two-Phase Flow of Incompressible Fluids with General Mass Densities

“…In [2], Abels, Depner, and Garcke prove existence of weak solutions for the case of logarithmic potentials; in [3], they consider mobilities M (ϕ) which degenerate in ϕ = ±1. In [19], Grün and Klingbeil suggest a numerical scheme for (1.1) which is discretely consistent with thermodynamics in the sense that in the absence of external forces the discrete counterpart of the total energy is decreasing in time.…”

“…Let us concentrate on the numerical aspects of diffuse interface models-for an overview of analytical results, we refer the reader to [1], [2], [3], and the references therein. Many authors have contributed to the numerics of diffuse interface models for two-phase flow in the special case that the two fluids share the same mass density.…”

Degenerate nonlocal Cahn-Hilliard equations: Well-posedness, regularity and local asymptotics

“…A nonlocal convective Cahn-Hilliard type system modelling phase-separation has been analyzed in [24,25]. Relevant studies in coupling the Cahn-Hilliard equation with a further equation for the velocity field have been the subject of [2,3,12,34]. From a mathematical viewpoint, the presence of convection terms (i.e.…”

Strong solutions to 1D compressible Navier‐Stokes/Allen‐Cahn system with free boundary

“…Meanwhile, noticing that the compressible case with free density is too difficult to deal with, some searchers have tried to deduce some different models basing on the assumption that the density depends on the concentration which satisfies Cahn‐Hilliard type equations and obtained some theoretical results, such as in the literature. ()…”

“…Meanwhile, noticing that the compressible case with free density is too difficult to deal with, some searchers have tried to deduce some different models basing on the assumption that the density depends on the concentration which satisfies Cahn-Hilliard type equations and obtained some theoretical results, such as in the literature. [13][14][15][16][17][18] Another type of diffuse interface model which has been extensively accepted and investigated is the coupled Navier-Stokes/Allen-Cahn system which has been proposed by Blesgen. 19 For incompressible fluids with matched density, Xu et al 20 proved the existence of axisymmetric solutions, Zhao et al 21 discussed the vanishing viscosity limit, and for the investigates on asymptotic behavior and attractors, see Gal and Grasselli.…”