Mathematical justification of a compressible bifluid system with different pressure laws: a continuous approach

ABSTRACT

This paper concerns the mathematical justification of a macroscopic Baer–Nunziato PDE bifluid system with a physical relaxation term that is linked to the two viscosities and the two pressure laws of the two compressible phases of the fluid which may be different. This is achieved using an homogenization approach in a periodic framework from a mesoscopic PDE description of two immiscible compressible viscous fluids with interfaces and no mass transfer. Our result extends the work in Bresch D, Hillairet M. [Note on the derivation of multi-component flow systems. Proc Am Math Soc. 2015;143:3429–3443] by allowing to consider different pressure laws for each component introducing an order parameter. This paper is complementary to the recent work [Bresch D, Burtea C, Lagoutière F. Mathematical justification of a compressible bi-fluid system with different pressure laws: a semi-discrete approach and numerical illustrations. Submitted 2021] which focuses on a semi-discretized approach and numerical illustrations. These two papers correspond to the extended versions of the document arXiv:2012.06497.

KEYWORDS:

• Compressible flows
• bifluid system
• Hoff solution
• homogenization
• Young measures
• physical relaxation

2020 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 76M50
• 76N06

Acknowledgements

The authors want to thank M. Hillairet for several discussions and comments.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

D. Bresch and C. Burtea are partially supported by SingFlows project, grant ANR-18-CE40-002. D. Bresch want to acknowledge support from MSRI for the research membership in the program ‘Mathematical problems in fluid dynamics’ organized by T. Alazard, H. Bahouri, M. Ifrim, I. Kukavica, D. Lannes, D. Tataru (spring 2021).