A Note on Hyperbolic Relaxation of the Navier-Stokes-Cahn-Hilliard system for incompressible two-phase flow

TL;DR

This paper introduces a hyperbolic relaxation approach to the Navier-Stokes-Cahn-Hilliard system for incompressible two-phase flow, enabling the use of hyperbolic numerical methods and providing initial numerical insights.

Contribution

It proposes a new first-order hyperbolic approximation of the Navier-Stokes-Cahn-Hilliard system using artificial compressibility and relaxation techniques.

Findings

The approximative system is hyperbolic under certain conditions.

Existence of an entropy-entropy flux pair for the 1D case.

Preliminary numerical results demonstrate the approach's potential.

Abstract

We consider the two-phase dynamics of two incompressible and immiscible fluids. As a mathematical model we rely on the Navier-Stokes-Cahn-Hilliard system that belongs to the class of diffuse-interface models. Solutions of the Navier-Stokes-Cahn-Hilliard system exhibit strong non-local effects due to the velocity divergence constraint and the fourth-order Cahn-Hilliard operator. We suggest a new first-order approximative system for the inviscid sub-system. It relies on the artificial-compressibility ansatz for the Navier-Stokes equations, a friction-type approximation for the Cahn-Hilliard equation and a relaxation of a third-order capillarity term. We show under reasonable assumptions that the first-order operator within the approximative system is hyperbolic; precisely we prove for the spatially one-dimensional case that it is equipped with an entropy-entropy flux pair with convex (mathematical) entropy. For specific states we present a numerical characteristic analysis. Thanks to the hyperbolicity of the system, we can employ all standard numerical methods from the field of hyperbolic conservation laws. We conclude the paper with preliminary numerical results in one spatial dimension.