| Issue |
ESAIM: M2AN
Volume 57, Number 5, September-October 2023
|
|
|---|---|---|
| Page(s) | 2865 - 2906 | |
| DOI | https://doi.org/10.1051/m2an/2023046 | |
| Published online | 19 September 2023 | |
Analysis of compressible bubbly flows. Part II: Derivation of a macroscopic model
1
Institut Montpelliérain Alexander Grothendieck, Univ Montpellier, CNRS, Montpellier, France
2
INRIA & Institut Montpelliérain Alexander Grothendieck, Univ Montpellier, CNRS, Montpellier, France
* Corresponding author: matthieu.hillairet@umontpellier.fr
Received:
15
September
2022
Accepted:
16
May
2023
This paper is the second of the series of two papers, which focuses on the derivation of an averaged 1D model for compressible bubbly flows. For this, we start from a microscopic description of the interactions between a large but finite number of small bubbles with a surrounding compressible fluid. This microscopic model has been derived and analysed in the first paper. In the present one, provided physical parameters scale according to the number of bubbles, we prove that solutions to the microscopic model exist on a timespan independent of the number of bubbles. Considering then that we have a large number of bubbles, we propose a construction of the macroscopic variables and derive the averaged system satisfied by these quantities. Our method is based on a compactness approach in a strong-solution setting. In the last section, we propose the derivation of the Williams–Boltzmann equation corresponding to our setting.
Mathematics Subject Classification: 76T05 / 76T10 / 35Q30
Key words: Homogenization / Two-phase flows / Compressible Navier–Stokes equations / Cauchy theory
© The authors. Published by EDP Sciences, SMAI 2023
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