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Structure-preserving, weighted implicit-explicit schemes for multi-phase incompressible Navier-Stokes/Darcy coupled nonlocal Allen-Cahn model

Meng Li, Ke Wang, Nan Wang

Abstract

A multitude of substances exist as mixtures comprising multiple chemical components in the natural world. These substances undergo morphological changes under external influences. the phase field model coupled with fluid flow, the dynamic movement and evolution of the phase interface intricately interact with the fluid motion. This article focuses on the N-component models that couple the conservative Allen-Cahn equation with two types of incompressible fluid flow systems: the Navier-Stokes equation and the Darcy equation. By utilizing the scalar auxiliary variable method and the projection method, we innovatively construct two types of structure-preserving weighted implicit-explicit schemes for the coupled models, resulting in fully decoupled linear systems and second-order accuracy in time. The schemes are proved to be mass-conservative. In addition, with the application of $G$-norm inspired by the idea of $G$-stability, we rigorously establish its unconditional energy stability. Finally, the performance of the proposed scheme is verified by some numerical simulations.

Structure-preserving, weighted implicit-explicit schemes for multi-phase incompressible Navier-Stokes/Darcy coupled nonlocal Allen-Cahn model

Abstract

A multitude of substances exist as mixtures comprising multiple chemical components in the natural world. These substances undergo morphological changes under external influences. the phase field model coupled with fluid flow, the dynamic movement and evolution of the phase interface intricately interact with the fluid motion. This article focuses on the N-component models that couple the conservative Allen-Cahn equation with two types of incompressible fluid flow systems: the Navier-Stokes equation and the Darcy equation. By utilizing the scalar auxiliary variable method and the projection method, we innovatively construct two types of structure-preserving weighted implicit-explicit schemes for the coupled models, resulting in fully decoupled linear systems and second-order accuracy in time. The schemes are proved to be mass-conservative. In addition, with the application of -norm inspired by the idea of -stability, we rigorously establish its unconditional energy stability. Finally, the performance of the proposed scheme is verified by some numerical simulations.
Paper Structure (18 sections, 9 theorems, 114 equations, 18 figures)

This paper contains 18 sections, 9 theorems, 114 equations, 18 figures.

Table of Contents

  1. Introduction
  2. Governing equations
  3. N-component NS-CAC model
  4. N-component D-CAC model
  5. Numerical schemes
  6. G-norm for the energy stability
  7. N-component NS-CAC model
  8. N-component D-CAC model
  9. Numerical experiments
  10. Convergence tests
  11. Energy dissipation and mass conservation
  12. 2-component models
  13. 3-component models
  14. The phase-separation of 3-component systems
  15. Conclusions
  16. ...and 3 more sections

Key Result

Lemma 3.1

\newlabellemma_GF-norm When $\theta \in [1/2,1]$, for any sequence of function $\{{\bf w}^n\}$, it holds

Figures (18)

  • Figure 4.1: Convergence tests of the 2-component NS-CAC model with external forcing terms
  • Figure 4.2: Convergence tests of the 2-component D-CAC model with external forcing terms
  • Figure 4.3: Convergence tests of the 3-component NS-CAC model with external forcing terms
  • Figure 4.4: Convergence tests of the 3-component D-CAC model with external forcing terms
  • Figure 4.5: The modified energy evolution of the 2-component NS-CAC model
  • ...and 13 more figures

Theorems & Definitions (22)

  • Remark 2.1
  • Remark 2.2
  • Lemma 3.1
  • Remark 3.2
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • proof
  • Remark 3.5
  • Remark 3.6
  • ...and 12 more