Justification of a Relaxation Approximation for the Navier-Stokes-Cahn-Hilliard System

TL;DR

This paper introduces a relaxation approximation for the Navier-Stokes-Cahn-Hilliard system, proves its solutions converge to the original system as relaxation parameters vanish, and provides numerical evidence using a novel finite-difference method.

Contribution

It develops a new relaxation approximation for the NSCH system, proves convergence to the original system, and demonstrates its effectiveness through numerical simulations of complex interfacial flows.

Findings

Solutions of the approximation recover the NSCH system as relaxation parameters tend to zero.

The approximate solutions dissipate an almost quadratic energy, aiding analysis.

Numerical simulations successfully model interfacial phenomena like Ostwald ripening and high-velocity flows.

Abstract

The Navier-Stokes-Cahn-Hilliard (NSCH) system governs the diffuse-interface dynamics of two incompressible and immiscible fluids. We consider a relaxation approximation of the NSCH system that is composed by a system of first-order hyperbolic balance laws and second-order elliptic operators. We prove first that the solutions of an initial boundary value problem for the approximation recover the limiting NSCH system for vanishing relaxation parameters. To cope with the singular limit we exploit the fact that the approximate solutions dissipate an almost quadratic energy, and employ the relative entropy-framework. In the second part of the work we provide numerical evidence for the analytical results, even in flow regimes not covered by the assumptions needed for the theoretical results. Using a novel marker-and-cell conservative finite-difference approach for both the approximation and the limit system, we are able to compute physically relevant interfacial flow problems including Ostwald ripening and high-velocity flow.