Investigation of isothermal Baer-Nunziato-type models

Siegfried Müller (RWTH Aachen University, Institut für Geometrie und Praktische Mathematik), Maren Hantke, Aleksey Sikstel, Ferdinand Thein

Compressible multi-component flows where the fluid is a mixture of several components all of which may be present in different aggregate states have a wide range of applications, for instance a mixture of reacting gases or a mixture of a liquid and a gas. They have been successfully modeled by Baer-Nunziato-type models, see for instance [4]. In recent years, work on barotropic Baer-Nunziatio models has been published in [3, 1, 5].

The main interest is on multi-component Baer-Nunziato-type models for mixtures with an arbitrary number of components all of which are modeled by immiscible isothermal fluids. These are given by balance equations for volume fractions, density and momentum for each component accounting for the relaxation to equilibrium by means of relaxation terms. In the talk the Baer-Nunziato model for isothermal flows is derived from the full Baer-Nunziato model, see [4], taking into account the heat flux.

Thermodynamical consisteny of the model is verified. For this purpose, first an appropriate Lax’ entropy-entropy flux pair is derived where, in particular, the phasic energy equations including the heat flux is accounted for, see [4]. Furthermore, to ensure an entropy equality appropriate interfacial pressures and interfacial velocity are chosen. In particular, unique interfacial pressures are determined depending on the interfacial velocity chosen as a convex combination of the phasic velocities. By this, the Baer-Nunziato models are closed.

Finally, instantaneous relaxation to equilibrium is investigated and appropriate algorithms are presented. These are used to perform numerical computations using a path-conservative DG scheme. Details on the presentation can be found in the preprint [2].

References

  1. Hamza Boukili, Jean-Marc Hérard. Relaxation and simulation of a barotropic three-phase flow model, 1031–1059, ESAIM: M2AN, 53(3), 2019. DOI 10.1051/m2an/2019001.

  2. M. Hantke, S. Müller, A. Sikstel, F. Thein. A survey on isothermal and isentropic Baer-Nunziato-type models, arXiv:2407.06919, July 2024.

  3. Jean-Marc Hérard. A class of compressible multiphase flow models, 954–959, Comptes Rendus Mathematique, 54(9), 2016. DOI 10.1016/j.crma.2016.07.004.

  4. S. Müller, M. Hantke, P. Richter. Closure conditions for non-equilibrium multi-component models, 1157–1189, Continuum Mechanics and Thermodynamics, 28(4), 2016. DOI 10.1007/s00161-015-0468-8.

  5. Khaled Saleh, Nicolas Seguin. Some mathematical properties of a barotropic multiphase flow model, 70–78, ESAIM: Proceedings, 60, 2020. DOI 10.1051/proc/202069070.