Fully Discrete Finite Element Approximations of the Navier--Stokes--Cahn-Hilliard Diffuse Interface Model for Two-Phase Fluid Flows

Abstract

This paper develops and analyzes some fully discrete finite element methods for a parabolic system consisting of the Navier--Stokes equations and the Cahn--Hilliard equation, which arises as a diffuse interface model for the flow of two immiscible and incompressible fluids. In the model the two sets of equations are coupled through an extra phase induced stress term in the Navier--Stokes equations and a fluid induced transport term in the Cahn--Hilliard equation. Fully discrete mixed finite element methods are proposed for approximating the coupled system, it is shown that the proposed numerical methods satisfy a mass conservation law, and a discrete energy law which is analogous to the basic energy law for the phase field model. The convergence of the numerical solutions to the solutions of the phase field model and its sharp interface limit is established by utilizing the discrete energy law. As a by-product, the convergence result also provides a constructive proof of the existence of weak solutions to the Navier--Stokes-Cahn--Hilliard phase field model. Numerical experiments are also presented to validate the theory and to show the effectiveness of the combined phase field and finite element approach.