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Nonisothermal Cahn-Hilliard Navier-Stokes system

Aaron Brunk, Dennis Schumann

TL;DR

This work develops a fully discrete, structure-preserving finite element scheme for the nonisothermal Cahn-Hilliard-Navier-Stokes (CHNST) system, ensuring mass conservation, total-energy conservation, and positive entropy production. It extends prior formulations by enabling unstable Navier–Stokes elements through Brezzi–Pitkäranta stabilization combined with Grad-Div stabilization, while incorporating stabilization terms into the internal-energy equation to maintain thermodynamic consistency. The method is analyzed and validated via convergence tests in 2D and three-dimensional simulations, demonstrating accurate energy and mass preservation and physically consistent entropy behavior, with notable efficiency gains over previous approaches. The results support robust simulations of coupled two-phase flow and heat transfer phenomena, with potential applications in additive manufacturing and related processes, and set the stage for further enhancements such as temperature-based discretizations and SUPG stabilization.

Abstract

In this research, we introduce and investigate an approximation method that preserves the structural integrity of the non-isothermal Cahn-Hilliard-Navier-Stokes system. Our approach extends a previously proposed technique [1], which utilizes conforming (inf-sup stable) finite elements in space, coupled with implicit time discretization employing convex-concave splitting. Expanding upon this method, we incorporate the unstable P1|P1 pair for the Navier-Stokes contributions, integrating Brezzi-Pitkäranta stabilization. Additionally, we improve the enforcement of incompressibility conditions through grad div stabilization. While these techniques are well-established for Navier-Stokes equations, it becomes apparent that for non-isothermal models, they introduce additional coupling terms to the equation governing internal energy. To ensure the conservation of total energy and maintain entropy production, these stabilization terms are appropriately integrated into the internal energy equation.

Nonisothermal Cahn-Hilliard Navier-Stokes system

TL;DR

This work develops a fully discrete, structure-preserving finite element scheme for the nonisothermal Cahn-Hilliard-Navier-Stokes (CHNST) system, ensuring mass conservation, total-energy conservation, and positive entropy production. It extends prior formulations by enabling unstable Navier–Stokes elements through Brezzi–Pitkäranta stabilization combined with Grad-Div stabilization, while incorporating stabilization terms into the internal-energy equation to maintain thermodynamic consistency. The method is analyzed and validated via convergence tests in 2D and three-dimensional simulations, demonstrating accurate energy and mass preservation and physically consistent entropy behavior, with notable efficiency gains over previous approaches. The results support robust simulations of coupled two-phase flow and heat transfer phenomena, with potential applications in additive manufacturing and related processes, and set the stage for further enhancements such as temperature-based discretizations and SUPG stabilization.

Abstract

In this research, we introduce and investigate an approximation method that preserves the structural integrity of the non-isothermal Cahn-Hilliard-Navier-Stokes system. Our approach extends a previously proposed technique [1], which utilizes conforming (inf-sup stable) finite elements in space, coupled with implicit time discretization employing convex-concave splitting. Expanding upon this method, we incorporate the unstable P1|P1 pair for the Navier-Stokes contributions, integrating Brezzi-Pitkäranta stabilization. Additionally, we improve the enforcement of incompressibility conditions through grad div stabilization. While these techniques are well-established for Navier-Stokes equations, it becomes apparent that for non-isothermal models, they introduce additional coupling terms to the equation governing internal energy. To ensure the conservation of total energy and maintain entropy production, these stabilization terms are appropriately integrated into the internal energy equation.
Paper Structure (6 sections, 3 theorems, 22 equations, 2 figures, 1 table)

This paper contains 6 sections, 3 theorems, 22 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Notation and Main Result
  3. Numerical Test
  4. Convergence test
  5. Three dimensional simulations
  6. Conclusion & Outlook

Key Result

lemma 1

A sufficiently regular solution $(\phi,\mu,\theta,\mathbf{u},p)$ of the system eq:ac1--eq:ac3 fulfills the variational formulation for sufficiently regular test functions $\psi,\xi,\mathbf{v},w,q$ and $\pi := p + e - \frac{s+\phi\mu}{\theta}$.

Figures (2)

  • Figure 1: Snapshots of the phase variable $\phi$ at the times $t\in\{0,0.03,0.06,0.1\}$.
  • Figure 2: Snapshots of the phase variable $\phi$ and the velocity field $\mathbf{u}$ at the times $t\in\{0,0.03,0.06,0.1\}$.

Theorems & Definitions (4)

  • lemma 1: brunk2024structurepreserving
  • theorem 1: brunk2024structurepreserving
  • theorem 2
  • proof