Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Regularized Framework and Admissible Solutions for Liquid-Vapor Phase Transitions in Steady Compressible Flows

Yazhou Chen, Qiaolin He, Dongjuan Niu, Yi Peng, Xiaoding Shi

TL;DR

The paper addresses liquid-vapor phase transitions in steady compressible flows described by the Navier–Stokes equations with a van der Waals equation of state under periodic boundary conditions. It introduces an artificial viscosity regularization and a variational framework to identify admissible, energy-minimizing phase-transition solutions, linking diffuse-interface models to Maxwell's construction. The main results show that the average specific volume $\bar{v}$ determines whether the flow is single-phase or exhibits a two-interface phase transition; as the viscosity vanishes, the diffuse interface sharpens to a Maxwell-discontinuous two-phase state, with the Maxwell region acting as a nucleation mechanism. The work provides a rigorous bridge between variational/phase-plane analyses and level-set methods to characterize and select physically meaningful two-phase states in isentropic van der Waals fluids.

Abstract

We investigate the well-posedness of the periodic boundary value problem for the steady compressible isentropic Navier-Stokes system under the van der Waals equation of state. The main difficulty arises from the non-monotonicity of the pressure, which induces liquid-vapor phase transitions and consequently leads to both physical instabilities and mathematical non-uniqueness of solutions. It is shown that the occurrence of a phase transition is determined by whether the integral average of the specific volume lies inside the gas-liquid coexistence region defined by the Maxwell construction. By introducing an artificial viscosity, we construct an approximate system. When the integral average of the specific volume falls within the Maxwell region, the approximate solution converges, as the artificial viscosity tends to zero, to the equilibrium states given by Maxwell's construction, with the diffuse interface sharpening into a discontinuity. Conversely, if the integral average of the specific volume lies outside this region, the limiting solution remains outside as well, meaning that no phase transition occurs. These results demonstrate that the non-monotonicity of the pressure, combined with the condition that the integral average of the specific volume belongs to the Maxwell region, can act as a nucleation mechanism for phase transitions in the isentropic gas-liquid problem. Furthermore, the proposed approximation not only offers a regularized framework for describing phase transitions but also provides, from a rigorous mathematical viewpoint, a definition of admissible solutions related to phase transitions. The detailed proof relies on the artificial viscosity method, the calculus of variations, the anti-derivative technique, phase-plane analysis, and the level-set method.

A Regularized Framework and Admissible Solutions for Liquid-Vapor Phase Transitions in Steady Compressible Flows

TL;DR

The paper addresses liquid-vapor phase transitions in steady compressible flows described by the Navier–Stokes equations with a van der Waals equation of state under periodic boundary conditions. It introduces an artificial viscosity regularization and a variational framework to identify admissible, energy-minimizing phase-transition solutions, linking diffuse-interface models to Maxwell's construction. The main results show that the average specific volume determines whether the flow is single-phase or exhibits a two-interface phase transition; as the viscosity vanishes, the diffuse interface sharpens to a Maxwell-discontinuous two-phase state, with the Maxwell region acting as a nucleation mechanism. The work provides a rigorous bridge between variational/phase-plane analyses and level-set methods to characterize and select physically meaningful two-phase states in isentropic van der Waals fluids.

Abstract

We investigate the well-posedness of the periodic boundary value problem for the steady compressible isentropic Navier-Stokes system under the van der Waals equation of state. The main difficulty arises from the non-monotonicity of the pressure, which induces liquid-vapor phase transitions and consequently leads to both physical instabilities and mathematical non-uniqueness of solutions. It is shown that the occurrence of a phase transition is determined by whether the integral average of the specific volume lies inside the gas-liquid coexistence region defined by the Maxwell construction. By introducing an artificial viscosity, we construct an approximate system. When the integral average of the specific volume falls within the Maxwell region, the approximate solution converges, as the artificial viscosity tends to zero, to the equilibrium states given by Maxwell's construction, with the diffuse interface sharpening into a discontinuity. Conversely, if the integral average of the specific volume lies outside this region, the limiting solution remains outside as well, meaning that no phase transition occurs. These results demonstrate that the non-monotonicity of the pressure, combined with the condition that the integral average of the specific volume belongs to the Maxwell region, can act as a nucleation mechanism for phase transitions in the isentropic gas-liquid problem. Furthermore, the proposed approximation not only offers a regularized framework for describing phase transitions but also provides, from a rigorous mathematical viewpoint, a definition of admissible solutions related to phase transitions. The detailed proof relies on the artificial viscosity method, the calculus of variations, the anti-derivative technique, phase-plane analysis, and the level-set method.
Paper Structure (5 sections, 20 theorems, 123 equations, 3 figures)

This paper contains 5 sections, 20 theorems, 123 equations, 3 figures.

Table of Contents

  1. Introduction and Main Results
  2. Proof of the Main Results
  3. Proof of Theorem 1.1.
  4. Proof of Theorem 1.2.
  5. Proof of Theorem 1.3.

Key Result

Lemma 1.1

(Properties of the pressure function.) Consider a subcritical temperature $\theta$, satisfying Under this condition, the pressure function $p(v)$ given by van der Waals equation exhibits the characteristic form shown in Figure 1 below:

Figures (3)

  • Figure 1: Figure of the pressure
  • Figure 2: phase transition solution with two distinct interfaces
  • Figure 3: image of $W(v)+\sigma v$ and phase plane of $v$--$v'$.

Theorems & Definitions (43)

  • Lemma 1.1
  • Remark 1.1
  • Remark 1.2
  • Theorem 1.1
  • Remark 1.3
  • Theorem 1.2
  • Remark 1.4
  • Definition 1.1
  • Remark 1.5
  • Definition 1.2
  • ...and 33 more