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Property-preserving numerical approximation of a Cahn-Hilliard-Navier-Stokes model with variable density and degenerate mobility

Daniel Acosta-Soba, Francisco Guillén-González, J. Rafael Rodríguez-Galván, Jin Wang

Abstract

In this paper, we present a new computational framework to approximate a Cahn-Hilliard-Navier-Stokes model with variable density and degenerate mobility that preserves the mass of the mixture, the pointwise bounds of the density and the decreasing energy. This numerical scheme is based on a finite element approximation for the Navier-Stokes fluid flow with discontinuous pressure and an upwind discontinuous Galerkin scheme for the Cahn-Hilliard part. Finally, several numerical experiments such as a convergence test and some well-known benchmark problems are conducted.

Property-preserving numerical approximation of a Cahn-Hilliard-Navier-Stokes model with variable density and degenerate mobility

Abstract

In this paper, we present a new computational framework to approximate a Cahn-Hilliard-Navier-Stokes model with variable density and degenerate mobility that preserves the mass of the mixture, the pointwise bounds of the density and the decreasing energy. This numerical scheme is based on a finite element approximation for the Navier-Stokes fluid flow with discontinuous pressure and an upwind discontinuous Galerkin scheme for the Cahn-Hilliard part. Finally, several numerical experiments such as a convergence test and some well-known benchmark problems are conducted.
Paper Structure (14 sections, 10 theorems, 58 equations, 7 figures, 2 tables)

This paper contains 14 sections, 10 theorems, 58 equations, 7 figures, 2 tables.

Table of Contents

  1. Keywords:
  2. Introduction
  3. Cahn--Hilliard--Navier--Stokes model
  4. Structure-preserving scheme
  5. Notation
  6. Discrete scheme
  7. Definition of the upwind bilinear form $\boldmath b_h^{\text{upw}}(\cdot;\cdot,\cdot)$
  8. Properties of the scheme \ref{['esquema_DG_NS-CH']}
  9. Numerical experiments
  10. Accuracy test
  11. Mixing bubbles
  12. A heavier bubble falling in a lighter medium
  13. A Rayleigh-Taylor type instability
  14. Conclusion

Key Result

Proposition 2.2

The mass of the phase-field variable is conserved, because it holds In particular, the mass of the mixture is conserved, because using rho,

Figures (7)

  • Figure 1: Initial condition of Tests \ref{['test:accuracy']} and \ref{['test:circle']}.
  • Figure 2: Evolution of $\Pi^h_1\phi$ over time in Test \ref{['test:circle']}.
  • Figure 3: Left, maximum and minimum of $\Pi^h_1\phi$. Right, discrete energy. Test \ref{['test:circle']}.
  • Figure 4: Evolution of $\Pi^h_1\phi$ over time in Test \ref{['test:bubble']}.
  • Figure 5: Left, maximum and minimum of $\Pi^h_1\phi$. Right, discrete energy. Test \ref{['test:bubble']}.
  • ...and 2 more figures

Theorems & Definitions (22)

  • Remark 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Proposition 3.4: Mass conservation
  • proof
  • ...and 12 more