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Constitutive Relations for Viscoleastic Fluid Models Derived from Kinetic Theory

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Dispersive Transport Equations and Multiscale Models

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 136))

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Abstract

Constitutive relations for the stress tensor in viscoselastic fluids are derived from the kinetic theory of polymeric liquids. The Fokker-Planck equation corresponding to the so-called dumbbells theory for diluted solutions of polymers is considered. The fluid models are derived for time-dependent, nonhomogeneous, and non-potential flows, thus extending the results of [2]. A comparison with Oldroyd-B, FENE and FENE-P fluids is presented in the frame of the plane Couette flow. This paper is an abriged version of [8].

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Refrences

  1. A. Bhave, R. Armstrong, AND R. Brown, Kinetic theory and rheology of dilute, nonhomogeneous polymer solutions, J. Chem. Phys., 95: 2988–3000 (1991).

    Article  Google Scholar 

  2. R. Bird, C. Curtiss, R. Armstrong, AND O. Hassager, Dynamics of polymeric liquids, Vol. 2, kinetic theory, John Wiley &, Sons, New-York, 1987.

    Google Scholar 

  3. J. Bonvin AND M. Picasso, Variance reduction methods for CONNFFESSIT-like simulations, J. Non-Newtonian Fluid Mech., 84: 191–215 (1999).

    Article  MATH  Google Scholar 

  4. H.J. Brascamp AND H. Lieb, On extensions of the Brunn-Minkowski and Précopa-Leindler theorems, including inequalities for log-concave functions, and with an application to the diffusion equation, Journal of Functional Analysis, 22: 366–389 (1976).

    Article  MathSciNet  MATH  Google Scholar 

  5. R. Caflish, The fluid-dynamic limit of the nonlinear Boltzmann equation, Comm. Pure Appl. Math., 33: 651–666 (1980).

    Article  MathSciNet  Google Scholar 

  6. C. Cercignani, R. Illner, AND M. Pulvirenti, The mathematical theory of dilute gases, Springer-Verlag, Berlin, 1994.

    MATH  Google Scholar 

  7. S. Chapman, The kinetic theory of simple and composite gases: viscosity, thermal conduction and diffusion, Proc. Roy. Soc. (London), 93: 1–20 (1916).

    Article  Google Scholar 

  8. P. Degond, M. Lemou, AND M. Picasso, Viscoleastic fluid models derived from kinetic equations for polymers, to appear in SIAM. Appl. Maths.

    Google Scholar 

  9. X. Gallez, P. Halin, G. Lielens, R. Keunings, AND V. Legat, The adaptive lagrangian particle method for macroscopic and micro?macro computations of time-dependent viscoelastic flows, Comp. Meth. Applied Mech. Engrg., 180: 345–364.

    Google Scholar 

  10. H. Grad, Principles of the kinetic theory of gases, Vol. 12 of Handbuch der Physik, Springer-Verlag, Berlin, 1958, pp. 205–294.

    Google Scholar 

  11. M. Herrchen AND H.-C. Öttinger, A detailed comparison of various FENE dumbbell models, J. Non-Newtonian Fluid Mech., 68: 17–42 (1997).

    Article  Google Scholar 

  12. M. Hulsen, A. VAN Heel, AND B. VAN DEN Brule, Simulation of viscoelastic flows using brownian configuration fields, J. Non-Newtonian Fluid Mech., 70: 79–101 (1997).

    Article  Google Scholar 

  13. L. Landau AND E. Lifshitz, Physical kinetics, in Theoretical Physics, Vol. 10, Akademie-Verlag, Berlin, 1983.

    Google Scholar 

  14. M. Laso AND H. Öttinger, Calculation of viscoelastic flow using molecular models, J. Non-Newtonian Fluid Mech., 47: 1–20 (1993).

    Article  MATH  Google Scholar 

  15. H.-C. Öttinger, Stochastic processes in polymeric fluids, Springer-Verlag, Berlin, 1996.

    Book  MATH  Google Scholar 

  16. M. Slemrod, Constitutive relations for monoatomic gases based on generalized rational approximation to the sum the chapman-enskog expansion, Arch. Rational Mech. Anal., 150: 1–22 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  17. M. Slemrod, Constitutive relations for Rivlin-Ericksen fluids based on generalized rational approximation, Arch. Rational Mech. Anal., 146: 73–93 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  18. A. VAN Heel, M. Hulsen, AND B. VAN DEN Brule, On the selection of parameters in the FENE-P model, J. Non-Newtonian Fluid Mech., 75: 253–271 (1998).

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Laboratoire MIP, Université Paul Sabatier, 118 route de Narbonne, 31062, Toulouse Cedex, France

    P. Degond & M. Lemou

  2. Département de Mathématiques, Ecole Polytechnique Fédérale de Lausanne, 1015, Lausanne, Switzerland

    M. Picasso

Authors
  1. P. Degond
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  2. M. Lemou
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  3. M. Picasso
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Editor information

Editors and Affiliations

  1. Laboratoire MIP, Université Paul Sabatier, Toulouse Cedex 4, 31062, France

    Naoufel Ben Abdallah  & Pierre Degond  & 

  2. Angewandte Mathematik, Universität des Saarlandes, Saarbrucken, D-66041, Germany

    Anton Arnold

  3. Department of Mathematics, University of Texas at Austin, Austin, TX, 78712, USA

    Irene M. Gamba

  4. Department of Mathematics, Indiana University, Bloomington, IN, 47405-5701, USA

    Robert T. Glassey

  5. CSCAMM, University of Maryland, College Park, MD, 20742-4015, USA

    C. David Levermore

  6. Department of Mathematics, Arizona State University, Tempe, AZ, 85287, USA

    Christian Ringhofer

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Degond, P., Lemou, M., Picasso, M. (2004). Constitutive Relations for Viscoleastic Fluid Models Derived from Kinetic Theory. In: Abdallah, N.B., et al. Dispersive Transport Equations and Multiscale Models. The IMA Volumes in Mathematics and its Applications, vol 136. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-8935-2_5

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  • DOI: https://doi.org/10.1007/978-1-4419-8935-2_5

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4612-6473-6

  • Online ISBN: 978-1-4419-8935-2

  • eBook Packages: Springer Book Archive

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