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

Jan Giesselmann, Jens Keim, Fabio Leotta, Christian Rohde

TL;DR

This work provides a rigorous convergence analysis of a low-order relaxation approximation to the Navier--Stokes--Cahn--Hilliard (NSCH) diffuse-interface model for incompressible two-phase flow. Leveraging a relative-energy framework, the authors prove that solutions of the relaxation system converge to the NSCH system as the relaxation parameters vanish, with explicit rates in the smooth regime. They introduce a novel MAC-based finite-difference method that discretizes both the NSCH system and its relaxation approximation, enabling stable computations in high-velocity and even inviscid settings. Numerical experiments in 1D and 2D—including Ostwald ripening, bubble evolution, and droplet merging/collision—confirm the theoretical convergence order and demonstrate robustness of the approach for physically relevant interfacial flows. Overall, the paper advances both analytical and numerical foundations for hyperbolic-relaxation approaches to diffuse-interface two-phase flow.

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.

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

TL;DR

This work provides a rigorous convergence analysis of a low-order relaxation approximation to the Navier--Stokes--Cahn--Hilliard (NSCH) diffuse-interface model for incompressible two-phase flow. Leveraging a relative-energy framework, the authors prove that solutions of the relaxation system converge to the NSCH system as the relaxation parameters vanish, with explicit rates in the smooth regime. They introduce a novel MAC-based finite-difference method that discretizes both the NSCH system and its relaxation approximation, enabling stable computations in high-velocity and even inviscid settings. Numerical experiments in 1D and 2D—including Ostwald ripening, bubble evolution, and droplet merging/collision—confirm the theoretical convergence order and demonstrate robustness of the approach for physically relevant interfacial flows. Overall, the paper advances both analytical and numerical foundations for hyperbolic-relaxation approaches to diffuse-interface two-phase flow.

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.
Paper Structure (34 sections, 8 theorems, 127 equations, 14 figures)

This paper contains 34 sections, 8 theorems, 127 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Phase-field modelling for incompressible two-phase flow
  3. The Navier--Stokes--Cahn--Hilliard system
  4. A low-order relaxation approximation of the NSCH system
  5. Convergence of the relaxation approximation
  6. The relative energy analysis
  7. Proof:
  8. Proof:
  9. Proof:
  10. Proof:
  11. Proof:
  12. Numerical justification of the relaxation approximation
  13. A numerical approach for the NSCH system and the relaxation approximation
  14. A finite-difference MAC method for the NSCH system
  15. A finite-difference MAC method for the relaxation approximation
  16. ...and 19 more sections

Key Result

Theorem 2.1

Let $(p,{\bf{u}},c)^T: \bar{\Omega}_T \to {\mathbb R}^{d+2}$ be a classical solution of the initial boundary value problem for (eq:NSCH) satisfying eq:iniNSCH, eq:boundNSCH. Then, we have for all $t \in [0,T]$ the energy balance relation

Figures (14)

  • Figure 1: Comparison of the spatial distribution of the solution variables for the NSCH system \ref{['eq:NSCH']} and its relaxation approximation\ref{['eq:NSCHr']}.
  • Figure 2: Left: Solution of the phase field variable at different time instances $t = 0$, $t = 0.15$, $t = 0.3$ with $N_x = 500$. Right: Temporal evolution of the total Helmholtz energy \ref{['eq:energy NSCH']} for different grid resolutions.
  • Figure 3: Convergence of the relaxation system \ref{['eq:NSCHr_num']} to the NSCH system \ref{['eq:NSCH']} for the Ostwald ripening test case with $N_x = 100$. In the top row, the convergence in terms of $\alpha$ for a fixed value of $\beta$ and varying $\delta$ is depicted. The center row shows the convergence in terms of $\beta$ for a fixed $\alpha$ and varying $\delta$. In the bottom row, the convergence error in terms of $\delta$ for a fixed value of $\alpha$ and varying $\delta$ is plotted. The red dashed lines indicate convergence with order $\mathcal{O}(\alpha)$, $\mathcal{O}(\beta)$ and $\mathcal{O}(\delta)$, depending on the respective running variable on the abscissa.
  • Figure 4: Temporal evolution of the total Helmholtz energy \ref{['eq:energy NSCH']} of the NSCH system \ref{['eq:NSCH']} for the single bubble, the merging droplets and the colliding droplets test cases, from left to right, each for three different grid resolutions.
  • Figure 5: Solution of the phase field variable for the 2D bubble test case at $t = 0$ (left), $t = 0.01$ (center) and $t = 0.25$ (right) computed with $N_x = N_y = 50$.
  • ...and 9 more figures

Theorems & Definitions (18)

  • Theorem 2.1: Energy dissipation for \ref{['eq:NSCH']}
  • Theorem 2.2: Energy dissipation for \ref{['eq:NSCHr']}
  • Definition 3.1: Classical solutions to the relaxation system with residuals
  • Definition 3.2: Weak solutions to the relaxation system
  • Proposition 3.3
  • Lemma 3.4: Growth rate of the full relative energy
  • Remark 3.5
  • Remark 3.6
  • Proposition 3.7
  • Lemma 3.8: Bound for the growth rate of the reduced relative energy
  • ...and 8 more