Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A fully discretization, unconditionally energy stable finite element method solving the thermodynamically consistent diffuse interface model for incompressible two-phase MHD flows with large density ratios

Ke Zhang

Abstract

A diffusion interface two-phase magnetohydrodynamic model has been used for matched densities in our previous work [1,2], which may limit the applications of the model. In this work, we derive a thermodynamically consistent diffuse interface model for diffusion interface two-phase magnetohydrodynamic fluids with large density ratios by Onsager's variational principle and conservation law for the first time. The finite element method for spatial discretization and the first order semi-implicit scheme linked with convect splitting method for temporal discretization, is proposed to solve this new model. The mass conservation, unconditionally energy stability and convergence of the scheme can be proved. Then we derive the existence of weak solutions of governing system employing the above properties of the scheme and compactness method. Finally, we show some numerical results to test the effectiveness and well behavior of proposed scheme.

A fully discretization, unconditionally energy stable finite element method solving the thermodynamically consistent diffuse interface model for incompressible two-phase MHD flows with large density ratios

Abstract

A diffusion interface two-phase magnetohydrodynamic model has been used for matched densities in our previous work [1,2], which may limit the applications of the model. In this work, we derive a thermodynamically consistent diffuse interface model for diffusion interface two-phase magnetohydrodynamic fluids with large density ratios by Onsager's variational principle and conservation law for the first time. The finite element method for spatial discretization and the first order semi-implicit scheme linked with convect splitting method for temporal discretization, is proposed to solve this new model. The mass conservation, unconditionally energy stability and convergence of the scheme can be proved. Then we derive the existence of weak solutions of governing system employing the above properties of the scheme and compactness method. Finally, we show some numerical results to test the effectiveness and well behavior of proposed scheme.
Paper Structure (13 sections, 9 theorems, 112 equations, 8 figures, 2 tables)

This paper contains 13 sections, 9 theorems, 112 equations, 8 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Model derivation and original energy
  3. Preliminaries
  4. Derivation of the model
  5. Fully discrete numerical scheme
  6. Description of the scheme
  7. Unconditionally energy stable of the scheme
  8. Existence of the solution of the scheme
  9. Convergence of the numerical scheme and existence of the weak solution
  10. Numerical examples
  11. Error test
  12. Spinodal decomposition
  13. Rising bubble

Key Result

Lemma 2.1

(Lax-Milgram Theorem) See for instance 2008, given a Hilbert space (V, ($\cdot,\cdot$)), a continuous, coercive bilinear form a($\cdot,\cdot$) and a continuous linear functional $\textbf{F}\in \textbf{V}'$, there exists a unique $\textbf{u}\in \textbf{V}$ such that

Figures (8)

  • Figure 1: Algorithm energy (a) and mass of phase field (b).
  • Figure 2: The dynamics of spinodal decomposition examples for scheme at t=0.0001 (a), 0.05 (b), 0.2 (c), 1 (d) with M=0.001.
  • Figure 3: The dynamics of spinodal decomposition examples for scheme at t=0.0001 (a), 0.05 (b), 0.2 (c), 1 (d) with M=1.
  • Figure 4: The phase evolution at t=0.1 (a), 1 (b), 2.5 (c), 3.5 (d) with R=0.15, $\frac{\rho_{2}}{\rho_{1}}$=9.
  • Figure 5: The phase evolution at t=0.1 (a), 1 (b), 2.5 (c), 3.5 (d) with R=0.2, $\frac{\rho_{2}}{\rho_{1}}$=9.
  • ...and 3 more figures

Theorems & Definitions (18)

  • Lemma 2.1
  • Remark 2.1
  • Theorem 2.1
  • proof
  • Definition 3.1
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Lemma 3.1
  • ...and 8 more