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Energetic variational approaches for multiphase flow systems with phase transition

Hajime Koba

TL;DR

The paper develops an energetic variational framework to derive and analyze mathematical models for multiphase flow with phase transition across a moving air–water interface. By partitioning the domain into an incompressible region $\Omega_A(t)$ and a compressible region $\Omega_B(t)$ with a constant interfacial density $\rho_0$ and surface tension $\pi_0$, it derives both viscous and inviscid two-phase systems, along with precise interface conditions and mass/conservation laws. The approach yields explicit stress constructs $\mathcal{T}_A$, $\mathcal{T}_B$, and their surface counterparts, enabling conservative forms and an energy identity that accounts for kinetic energy, viscous dissipation, and surface-energy fluxes; Euler/Navier–Stokes regimes arise as special cases. This variational formulation provides a rigorous, adaptable framework for modeling air–sea interactions and phase-transition phenomena in multiphase flows.

Abstract

We study the governing equations for the motion of the fluid particles near air-water interface from an energetic point of view. Since evaporation and condensation phenomena occur at the interface, we have to consider phase transition. This paper applies an energetic variational approach to derive multiphase flow systems with phase transition, where a multiphase flow means compressible and incompressible two-phase flow. We also research the conservation and energy laws of our system. The key ideas of deriving our systems are to acknowledge the existence of the interface and to apply an energetic variational approach. More precisely, we assume that both the coefficient of surface tension and the density of the interface are constants, and we apply an energetic variational approach to look for the dominant equations for the densities of our multiphase flow systems with phase transition. As applications, we can derive the usual Euler and Navier-Stokes systems, or a two-phase flow system with surface tension by our methods.

Energetic variational approaches for multiphase flow systems with phase transition

TL;DR

The paper develops an energetic variational framework to derive and analyze mathematical models for multiphase flow with phase transition across a moving air–water interface. By partitioning the domain into an incompressible region and a compressible region with a constant interfacial density and surface tension , it derives both viscous and inviscid two-phase systems, along with precise interface conditions and mass/conservation laws. The approach yields explicit stress constructs , , and their surface counterparts, enabling conservative forms and an energy identity that accounts for kinetic energy, viscous dissipation, and surface-energy fluxes; Euler/Navier–Stokes regimes arise as special cases. This variational formulation provides a rigorous, adaptable framework for modeling air–sea interactions and phase-transition phenomena in multiphase flows.

Abstract

We study the governing equations for the motion of the fluid particles near air-water interface from an energetic point of view. Since evaporation and condensation phenomena occur at the interface, we have to consider phase transition. This paper applies an energetic variational approach to derive multiphase flow systems with phase transition, where a multiphase flow means compressible and incompressible two-phase flow. We also research the conservation and energy laws of our system. The key ideas of deriving our systems are to acknowledge the existence of the interface and to apply an energetic variational approach. More precisely, we assume that both the coefficient of surface tension and the density of the interface are constants, and we apply an energetic variational approach to look for the dominant equations for the densities of our multiphase flow systems with phase transition. As applications, we can derive the usual Euler and Navier-Stokes systems, or a two-phase flow system with surface tension by our methods.
Paper Structure (8 sections, 7 theorems, 74 equations, 1 figure)

This paper contains 8 sections, 7 theorems, 74 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main Results
  3. Laws of Conservation of Mass
  4. Mathematical Modeling
  5. Viscous Model
  6. Inviscid Model
  7. Conservation and Energy Laws
  8. Appendix: Tools

Key Result

Theorem 2.2

$(\rm{i})$ Assume that $(\rho_A, \rho_B, \rho_0, v_A, v_B, v_S, \pi_A, \pi_B,\pi_0)$ satisfy and eq12. Then eq17 holds for all $0 < t_1 < t_2 <T$. $(\rm{ii})$ Assume that $(\rho_A, \rho_B, \rho_0, v_A, v_B, v_S, \pi_A, \pi_B,\pi_0, \mu_A, \mu_B, \lambda_B)$ satisfy and eq15, where $(\mathcal{T}_A, \mathcal{T}_B, \widetilde{\mathcal{T}}_A, \widetilde{\mathcal{T}}_B)$ are defined by eq16. Then eq1

Figures (1)

  • Figure 1: Multiphase Flow and Phase Transition

Theorems & Definitions (13)

  • Remark 1.1
  • Definition 2.1: $\Omega_T$ is flowed by the velocity fields $(v_A,v_B,v_S)$
  • Theorem 2.2: Laws of conservation of mass
  • Theorem 2.3: Conservative form, conservation and energy Laws
  • proof : Proof of Theorem \ref{['thm22']}
  • Lemma 3.1
  • proof : Proof of Lemma \ref{['lem31']}
  • Remark 4.1
  • Lemma 4.2
  • proof : Proof of Lemma \ref{['lem42']}
  • ...and 3 more