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A Variational Formulation for Fluid Dynamics with Irreversible Processes

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Geometric Science of Information (GSI 2017)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 10589))

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Abstract

In this paper, we present a variational formulation for heat conducting viscous fluids, which extends the Hamilton principle of continuum mechanics to include irreversible processes. This formulation follows from the general variational description of nonequilibrium thermodynamics introduced in [3, 4] for discrete and continuum systems. It relies on the concept of thermodynamic displacement. The irreversibility is encoded into a nonlinear nonholonomic constraint given by the expression of the entropy production associated to the irreversible processes involved.

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Notes

  1. 1.

    The notion of thermal displacement was first used by [13] and in the continuum setting by [9]. We refer to the Appendix of [11] for an historical account.

  2. 2.

    In this paper we do not describe the functional analytic setting needed to rigorously work in the framework of infinite dimensional manifolds. For example, one can assume that the diffeomorphisms are of some given Sobolev class, regular enough (at least of class \(C ^1\)), so that \({\text {Diff}}_0( \mathcal {D} )\) is a smooth infinite dimensional manifold and a topological group with smooth right translation, [1].

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Acknowledgements

F.G.B. is partially supported by the ANR project GEOMFLUID, ANR-14-CE23-0002-01; H.Y. is partially supported by JSPS Grant-in-Aid for Scientific Research (26400408, 16KT0024, 24224004), Waseda University (SR2017K-167), and the MEXT “Top Global University Project”.

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Authors and Affiliations

  1. CNRS & Ecole Normale Supérieure, LMD, IPSL, 24 Rue Lhomond, 75005, Paris, France

    François Gay-Balmaz

  2. School of Science and Engineering, Waseda University, Okubo, Shinjuku, Tokyo, 169-8555, Japan

    Hiroaki Yoshimura

Authors
  1. François Gay-Balmaz
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  2. Hiroaki Yoshimura
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Correspondence to François Gay-Balmaz .

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Editors and Affiliations

  1. Ecole Polytechnique, Palaiseau, France

    Frank Nielsen

  2. Thales Land and Air Systems, Limours, France

    Frédéric Barbaresco

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Gay-Balmaz, F., Yoshimura, H. (2017). A Variational Formulation for Fluid Dynamics with Irreversible Processes. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2017. Lecture Notes in Computer Science(), vol 10589. Springer, Cham. https://doi.org/10.1007/978-3-319-68445-1_47

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  • DOI: https://doi.org/10.1007/978-3-319-68445-1_47

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