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Low-Mach-number limit of a compressible two-phase flow system with algebraic closure

Cassandre Lebot

Abstract

We analyse a bi-fluid isentropic compressible Navier-Stokes system with barotropic pressure laws in a two-phase framework with equal pressure and single velocity. We focus on the rigorous analysis of the low Mach number limit under well-prepared initial data. Our main result shows that, as the Mach number tends to zero, the partial densities converge to constant states while the velocity field converges to a divergence-free vector field, and we recover the incompressible non-homogenous fluid system. The volume fractions remain nontrivial and are transported by the limit flow. Our method is based on the introduction of suitable modulated quantities and a novel relative entropy functional adapted to the two-phase structure. The key novelty lies in an original comparison of L1 and L2 norms of the partial densities, exploiting the special structure of the bi-fluid system, together with a variant of Csiszar-Kullback- Pinsker inequality. It allows us to rigorously justify the convergence of weak solutions of the compressible system toward solutions of the corresponding incompressible limit system in the low Mach number regime.

Low-Mach-number limit of a compressible two-phase flow system with algebraic closure

Abstract

We analyse a bi-fluid isentropic compressible Navier-Stokes system with barotropic pressure laws in a two-phase framework with equal pressure and single velocity. We focus on the rigorous analysis of the low Mach number limit under well-prepared initial data. Our main result shows that, as the Mach number tends to zero, the partial densities converge to constant states while the velocity field converges to a divergence-free vector field, and we recover the incompressible non-homogenous fluid system. The volume fractions remain nontrivial and are transported by the limit flow. Our method is based on the introduction of suitable modulated quantities and a novel relative entropy functional adapted to the two-phase structure. The key novelty lies in an original comparison of L1 and L2 norms of the partial densities, exploiting the special structure of the bi-fluid system, together with a variant of Csiszar-Kullback- Pinsker inequality. It allows us to rigorously justify the convergence of weak solutions of the compressible system toward solutions of the corresponding incompressible limit system in the low Mach number regime.
Paper Structure (18 sections, 11 theorems, 137 equations)

This paper contains 18 sections, 11 theorems, 137 equations.

Table of Contents

  1. Introduction
  2. Bi-fluid compressible system with algebraic closure
  3. Bi-fluid incompressible system
  4. Main theorem
  5. Relative entropy
  6. Definition
  7. First part of the entropy
  8. Second part of the entropy
  9. Key estimates for the convergence analysis
  10. A key $L^2$-estimate for the partial densities
  11. Csiszár-Kullback-Pinsker-type inequality
  12. Strong convergence of the partial densities $R_\pm^\varepsilon$
  13. Uniform bound for ${\mathcal{E}}_2$.
  14. Uniform bound for ${\mathcal{E}}_1$.
  15. Uniform bound for ${\mathcal{E}}$
  16. ...and 3 more sections

Key Result

Theorem 1.3

Let $(\rho_0, u_0)$ be some initial data such that $u_0 \in H^1_0(\Omega) \cap H^s(\Omega)$, $\operatorname{div} u_0 = 0$ in $\Omega$ and $\rho_0 \in L^\infty(\Omega)$, with $\overline{C} \geqslant \rho_0 \geqslant \underline{C}>0$ for some constants $\overline{C}, \underline{C}$. Then there exists

Theorems & Definitions (28)

  • Remark 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.6: Singular limit of the compressible two-phase system
  • Remark 1.7
  • Remark 1.8
  • Definition 2.1
  • Remark 2.2
  • ...and 18 more