Weak solutions and incompressible limit of a quasi-incompressible Navier--Stokes/Cahn--Hilliard model for viscous two-phase flows

TL;DR

This paper proves the existence of weak solutions for a quasi-incompressible Navier--Stokes/Cahn--Hilliard model describing two-phase flows with unmatched densities and long-range interactions, and establishes the incompressible limit as density differences vanish.

Contribution

It introduces new regularity estimates and pressure controls, and rigorously demonstrates the convergence to the incompressible model in a three-dimensional setting.

Findings

Existence of global weak solutions in 3D periodic domains.

New regularity estimate for the order parameter.

Convergence to the incompressible model as density difference tends to zero.

Abstract

We study a quasi-incompressible Navier--Stokes/Cahn--Hilliard coupled system which describes the motion of two macroscopically immiscible incompressible viscous fluids with partial mixing in a small interfacial region and long-range interactions. The case of unmatched densities with mass-averaged velocity is considered so that the velocity field is no longer divergence-free, and the pressure enters the equation of the chemical potential. We first prove the existence of global weak solutions to the model in a three-dimensional periodic domain, for which the implicit time discretization together with a fixed-point argument to the approximate system is employed. In particular, we obtain a new regularity estimate of the order parameter by exploiting the partial damping effect of the capillary force. Then utilizing the relative entropy method, we establish the incompressible limit -- the quasi-incompressible two-phase model converges to model H as the density difference tends to zero. Crucial to the passage of the incompressible limit, due to the lack of regularity of the pressure, are some non-standard uniform-in-density difference controls of the pressure, which are derived from the structure of the momentum equations and the improved regularity of the order parameter.