Explicit model for a mixture of Eulerian fluids based on kinetic theory
Author
Milana Colic
Title
Explicit model for a mixture of Eulerian fluids based on kinetic theory
Description
Supplementary material
Category
Working Material
Keywords
Boltzmann operator, multi-velocity and multi-temperature model, phenomenological coefficients, gas mixtures, polyatomic gases
URL
http://www.notebookarchive.org/2024-05-97bv7am/
DOI
https://notebookarchive.org/2024-05-97bv7am
Date Added
2024-05-20
Date Last Modified
2024-05-20
File Size
138.7 kilobytes
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Rights
CC BY-SA 4.0
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Supplementary material for the paperExplicit model for a mixture of Eulerian fluids based on kinetic theoryby Milana ČolićDepartment of Mathematics and Informatics, University of Novi Sad, Serbia & Applied and Computational Mathematics, RWTH Aachen University, Germanymilana.colic@dmi.uns.ac.rsAbstract of the paper. Within extended thermodynamics, a mixture of non-reactive Eulerian fluids is described with the multi-velocity and multi-temperature system of Euler-like equations, consisting in conservation laws for mass densities and balance laws for momentum and energy densities of each mixture component. The source terms are of phenomenological nature and appear in the system due to the mutual interaction between species. The aim of this paper is to derive such a macroscopic model starting from the system of Boltzmann-like equations describing a mixture of both monatomic and polyatomic gases. The method relies on an exact maximum entropy principle and evaluation of the Boltzmann multi-species collision operators for the collision kernel used in a recent rigorous mathematical analysis. Based on the obtained highly non-linear momentum and energy production terms, we propose a non-linear closure for the phenomenological model in the case of multi-velocity/single-temperature and single-velocity/multi-temperature. For the multi-velocity and multi-temperature case, we propose a closure via linearization in the neighborhood of the mixture mean velocity and temperature.This notebook implements the algorithm to compute production terms in the balance laws for momentum and energy densities of each mixture component. The design of the algorithm is presented in the Appendix D of the paper mentioned above and is based on the evaluation of the Boltzmann multi-species collision operator for a binary interaction between two molecules, either monatomic or polyatomic.The method is inspired by a polyatomic single-component case:Vladimir Djordjic, Milana Pavic-Colic, Manuel Torrilhon, “Boltzmann Collision Operator for Polyatomic Gases” from the Notebook Archive (2021), https://notebookarchive.org/2021-05-607ilfoThe notebook is organized as follows. Chapter 1 introduces assumptions and notation. Chapter 2 presents the evaluation procedure of poly-poly Boltzmann collision operator. Chapter 3 states the final results of the evaluation of all four possible combinations of Boltzmann collision operators in a multi-component mixture of monatomic and polyatomic gases (mono-mono, mono-poly, poly-mono, poly-poly). More precisely, referring to the paper Explicit model for a mixture of Eulerian fluids based on kinetic theory mentioned above, the coefficients and from Proposition 2, listed in Appendix B3 & B4, are typed in Chapter 3 of this notebook. Moreover, coefficients in momentum production terms in a multi-velocity/single temperature model (Appendix B5) and energy production terms in a single-velocity/multi-temperature model (Appendix B6) are typed, together with the coefficients of the linearized multi-velocity/multi-temperature model (Appendix B7 & B8).
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Chapter 1. Assumptions and Notation
Assumptions Run Section
Assumptions
Run Section
Notation
Notation
Constants Run Section
Constants
Run Section
Notation for hypergeometric functions (run only when using Chapter 3) Run Section
Notation for hypergeometric functions (run only when using Chapter 3)
Run Section
Chapter 2. Evaluation of the Boltzmann multi-species collision operator for Eulerian fluids
Integration formulas Run Section
Integration formulas
Run Section
Evaluation of the poly-poly Boltzmann collision integral
Evaluation of the poly-poly Boltzmann collision integral
Chapter 3. Final expressions for coefficients in the production terms given in Appendix B
Coefficients in the multi-velocity and multi-temperature model (Appendix B3 & B4)
Coefficients in the multi-velocity and multi-temperature model (Appendix B3 & B4)
Momentum production term coefficient in the multi-velocity and single-temperature model (Appendix B5)
Momentum production term coefficient in the multi-velocity and single-temperature model (Appendix B5)
Energy production term coefficient in the single-velocity and multi-temperature model (Appendix B6)
Energy production term coefficient in the single-velocity and multi-temperature model (Appendix B6)
Coefficients in the linearized multi-velocity and multi-temperature model (Appendix B7 & B8)
Coefficients in the linearized multi-velocity and multi-temperature model (Appendix B7 & B8)
Cite this as: Milana Colic, "Explicit model for a mixture of Eulerian fluids based on kinetic theory" from the Notebook Archive (2024), https://notebookarchive.org/2024-05-97bv7am
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