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Two-phase averaged system justification for ideal gases without conductivity

D Bresch, C Burtea, P Gonin--Joubert, F Lagoutière

TL;DR

The paper tackles the rigorous derivation of a two-phase averaged model for a one-dimensional compressible viscous fluid without heat conduction, addressing oscillations in density and temperature that persist in the limit. It develops a Hoff-type framework for a mesoscopic two-fluid system with a transported color function, then uses homogenization and Young measures to obtain a macroscopic two-phase model with two temperatures. A key contribution is establishing strong convergence of the stress tensor and deriving a kinetic equation for the associated Young measures, enabling a closed macroscopic description akin to Baer–Nunziato-type models. Numerical simulations illustrate the convergence between mesoscopic oscillatory solutions and the macroscopic two-phase system, confirming the theoretical results and highlighting Hoff-type discontinuities and their temporal decay.

Abstract

This article concerns the mathematical justification of an averaged system of partial differential equations governing the evolution of a two-phase mixture of compressible ideal fluids, with viscosity and without conductivity, in space dimension 1 with periodic boundary conditions. The derivation is done by some homogenization procedure. The originality and the difficulty of the paper consists in the fact that both the density and temperature are allowed to oscillate (because of the absence of heat conduction), so that the limiting model is a six-equations, two-pressures, two-temperatures model. The key point is to show the strong convergence of the stress tensor in $\(L^2((0,T)\times (0, 1))\)$. The main difficulties are to obtain uniform estimates in spite of the presence of oscillating coefficients in the energy equation. It requires to look at solutions with low regularity for the density and the temperature.

Two-phase averaged system justification for ideal gases without conductivity

TL;DR

The paper tackles the rigorous derivation of a two-phase averaged model for a one-dimensional compressible viscous fluid without heat conduction, addressing oscillations in density and temperature that persist in the limit. It develops a Hoff-type framework for a mesoscopic two-fluid system with a transported color function, then uses homogenization and Young measures to obtain a macroscopic two-phase model with two temperatures. A key contribution is establishing strong convergence of the stress tensor and deriving a kinetic equation for the associated Young measures, enabling a closed macroscopic description akin to Baer–Nunziato-type models. Numerical simulations illustrate the convergence between mesoscopic oscillatory solutions and the macroscopic two-phase system, confirming the theoretical results and highlighting Hoff-type discontinuities and their temporal decay.

Abstract

This article concerns the mathematical justification of an averaged system of partial differential equations governing the evolution of a two-phase mixture of compressible ideal fluids, with viscosity and without conductivity, in space dimension 1 with periodic boundary conditions. The derivation is done by some homogenization procedure. The originality and the difficulty of the paper consists in the fact that both the density and temperature are allowed to oscillate (because of the absence of heat conduction), so that the limiting model is a six-equations, two-pressures, two-temperatures model. The key point is to show the strong convergence of the stress tensor in . The main difficulties are to obtain uniform estimates in spite of the presence of oscillating coefficients in the energy equation. It requires to look at solutions with low regularity for the density and the temperature.
Paper Structure (21 sections, 22 theorems, 269 equations, 1 figure)

This paper contains 21 sections, 22 theorems, 269 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Modeling and Formal Derivation
  3. Compressible Navier--Stokes equations with temperature
  4. Mesoscopic unmixed phases equations
  5. Macroscopic averaged equations: Formal derivation
  6. Formal statement of the main results
  7. Global solution and uniqueness for Navier-Stokes with low regularity temperature
  8. Energy estimates and bounds
  9. Notations and properties of operators
  10. Conservative quantities and bounds related to rho, theta, rho theta and sigma.
  11. Smooth solutions and stability
  12. Uniqueness
  13. Averaging procedure
  14. Convergence of the unknowns using the uniform bounds
  15. Kinetic equation
  16. ...and 6 more sections

Key Result

Proposition 2.1

Let us assume that $\sigma^\varepsilon$ and $u^\varepsilon$ strongly converges to $\sigma$ and $u$ in $L^2(0,T;L^2(\mathbb{T}))$. Moreover let us consider that there exists $\alpha_\pm \in [0,1]$, $\rho_\pm\in [\underline\rho, \overline\rho]$, $\theta_\pm\in [\underline\theta, \overline\theta]$ som Assuming where namely then where $e_\pm=\mathop{\mathrm{\mathfrak{c}_v}}\nolimits_\pm \theta_\p

Figures (1)

  • Figure 1: Error with respect to the number of cells $J$

Theorems & Definitions (51)

  • Remark 2.1
  • Proposition 2.1
  • Remark 2.2
  • Remark 2.3
  • proof
  • Proposition 2.2
  • proof
  • Corollary 2.1
  • proof
  • Remark 2.4
  • ...and 41 more