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Handbook of Mathematical Analysis in Mechanics of Viscous Fluids pp 1–41Cite as

Variational Modeling and Complex Fluids

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Abstract

In this chapter, a general energetic variational framework for modeling the dynamics of complex fluids is introduced. The approach reveals and focuses on the couplings and competitions between different mechanisms involved for specific materials, including energetic contributions vs. kinematic transport relations, conservative parts vs. dissipative parts and kinetic parts vs. free energy parts of the systems, macroscopic deformation or flows vs. microscopic deformations, bulk effects vs. boundary conditions, etc. One has to notice that these variational approaches are motivated by the seminal works of Rayleigh (Proc Lond Math Soc 1(1):357–368, 1871) and Onsager (Phys Rev 37(4):405, 1931; Phys Rev 38(12):2265, 1931). In this chapter, the underlying physical principles and background, as well as the limitations of these approaches, are demonstrated. Besides the classical models for ideal fluids and elastic solids, these approaches are employed for models of viscoelastic fluids, diffusion, and mixtures.

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Acknowledgements

The work of Arkadz Kirshtein and Chun Liu is partially supported by NSF grants DMS-141200 and DMS-1216938. Authors want to thank Professor Yoshikazu Giga for reading the earlier draft of the manuscript, helpful discussions, and suggestions.

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Authors and Affiliations

  1. Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba Meguro-ku, 153-8914, Tokyo, Japan

    Mi-Ho Giga

  2. Department of Mathematics, Pennsylvania State University, McAllister Bldg, 16802, University Park, PA, USA

    Arkadz Kirshtein

  3. Department of Mathematics, Pennsylvania State University, McAllister Bldg, 16802, University Park, PA, USA

    Chun Liu

Authors
  1. Mi-Ho Giga
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  2. Arkadz Kirshtein
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  3. Chun Liu
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Correspondence to Arkadz Kirshtein .

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Editors and Affiliations

  1. Tokyo, Tokyo, Japan

    Yoshikazu Giga

  2. Toulon, Var, France

    Antonin Novotny

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Giga, MH., Kirshtein, A., Liu, C. (2017). Variational Modeling and Complex Fluids. In: Giga, Y., Novotny, A. (eds) Handbook of Mathematical Analysis in Mechanics of Viscous Fluids. Springer, Cham. https://doi.org/10.1007/978-3-319-10151-4_2-1

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  • DOI: https://doi.org/10.1007/978-3-319-10151-4_2-1

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-10151-4

  • Online ISBN: 978-3-319-10151-4

  • eBook Packages: Springer Reference MathematicsReference Module Computer Science and Engineering

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