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Effective Equations for a Compressible Liquid-Vapor Flow Model with Highly Oscillating Initial Density

Christian Rohde, Florian Wendt

TL;DR

Addresses homogenization of a 3D compressible liquid-vapor flow with many phase boundaries by replacing the detailed NSK dynamics with a parabolic relaxation (PNSK) and interpreting density oscillations as a Young measure. Proves convergence to an effective system comprising a deterministic hydrodynamic part (velocity and order parameter) and a kinetic equation for the limit measure, which is recast into a kinetic equation for the cumulative distribution function. The result yields a closed, computable framework for spray-like two-phase flows, linking micro-scale phase mixing to macro-scale observables, and extends previous 1D/constant-viscosity homogenization to 3D with phase-transition effects. The work lays groundwork for practical approximations and further extensions, including non-isothermal and nonlocal NSK variants.

Abstract

We derive and justify a new effective model for a compressible viscous liquid-vapor flow on a spray-like scale, i.e., for settings with a large number of phase boundaries. As a model on the detailed scale, we start from a parabolic relaxation of the Navier-Stokes-Korteweg system. We consider a sequence of initial data where the sequence of initial densities is assumed to be highly oscillating mimicking the high number of phase boundaries initially. Then, we consider a sequence of finite energy weak solutions corresponding to the sequence of initial data. Anticipating that the effective equations are found in the limit of infinitely many initial phase changes, we interpret the densities as Young measures and prove the convergence of the sequence of solutions to the effective model. The effective model consists of a deterministic part for the fluid's hydrodynamic quantities and a kinetic equation for the limit Young measure encoding the mixing dynamics. By characterizing the Young measure with the corresponding cumulative distribution function, we rewrite the kinetic equation for the Young measure into a kinetic equation for the cumulative distribution function such that the resulting equations are accessible by standard approximation methods.

Effective Equations for a Compressible Liquid-Vapor Flow Model with Highly Oscillating Initial Density

TL;DR

Addresses homogenization of a 3D compressible liquid-vapor flow with many phase boundaries by replacing the detailed NSK dynamics with a parabolic relaxation (PNSK) and interpreting density oscillations as a Young measure. Proves convergence to an effective system comprising a deterministic hydrodynamic part (velocity and order parameter) and a kinetic equation for the limit measure, which is recast into a kinetic equation for the cumulative distribution function. The result yields a closed, computable framework for spray-like two-phase flows, linking micro-scale phase mixing to macro-scale observables, and extends previous 1D/constant-viscosity homogenization to 3D with phase-transition effects. The work lays groundwork for practical approximations and further extensions, including non-isothermal and nonlocal NSK variants.

Abstract

We derive and justify a new effective model for a compressible viscous liquid-vapor flow on a spray-like scale, i.e., for settings with a large number of phase boundaries. As a model on the detailed scale, we start from a parabolic relaxation of the Navier-Stokes-Korteweg system. We consider a sequence of initial data where the sequence of initial densities is assumed to be highly oscillating mimicking the high number of phase boundaries initially. Then, we consider a sequence of finite energy weak solutions corresponding to the sequence of initial data. Anticipating that the effective equations are found in the limit of infinitely many initial phase changes, we interpret the densities as Young measures and prove the convergence of the sequence of solutions to the effective model. The effective model consists of a deterministic part for the fluid's hydrodynamic quantities and a kinetic equation for the limit Young measure encoding the mixing dynamics. By characterizing the Young measure with the corresponding cumulative distribution function, we rewrite the kinetic equation for the Young measure into a kinetic equation for the cumulative distribution function such that the resulting equations are accessible by standard approximation methods.

Paper Structure

This paper contains 7 sections, 16 theorems, 217 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Admissible Pressure Functions and Weak Solutions
  3. Main Result
  4. Uniform Estimates
  5. Proof of the Main Theorem
  6. Conclusions
  7. Compactness Results and Weak Continuity in Time

Key Result

Theorem 3.2

Let $T,\mu,\lambda,\kappa,\alpha,\beta>0$ and let $\Omega \subseteq \mathbb{R}^3$ be a bounded domain with regular boundary. Assume that $P$ is admissible with growth rate $\gamma \in (1,\infty)$ and let ${\tilde{\gamma}}:=\max\{2,\gamma\}$. For $n \in \mathbb{N}$, let initial data be given such that and suppose that there exists a Young measure $\nu^0\in \mathcal{Y}(\Omega;\mathcal{M}(\mathbb{R

Figures (1)

  • Figure 1: Example of a pressure function of Van-der-Waals type. Left: the original pressure function $P$. Right: the corresponding artificial pressure function $P_\alpha$ for a fixed coupling coefficient $\alpha>0$.

Theorems & Definitions (36)

  • Definition 2.1: Admissible pressure function
  • Remark 2.2
  • Definition 2.3: Finite energy weak solutions
  • Definition 3.1: Young measure
  • Theorem 3.2
  • Corollary 3.3
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • ...and 26 more