Two shallow-water type models for viscoelastic flows from kinetic theory for polymers solutions

Gladys Narbona-Reina; Didier Bresch

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2013)

  • Volume: 47, Issue: 6, page 1627-1655
  • ISSN: 0764-583X

Abstract

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In this work, depending on the relation between the Deborah, the Reynolds and the aspect ratio numbers, we formally derived shallow-water type systems starting from a micro-macro description for non-Newtonian fluids in a thin domain governed by an elastic dumbbell type model with a slip boundary condition at the bottom. The result has been announced by the authors in [G. Narbona-Reina, D. Bresch, Numer. Math. and Advanced Appl. Springer Verlag (2010)] and in the present paper, we provide a self-contained description, complete formal derivations and various numerical computations. In particular, we extend to FENE type systems the derivation of shallow-water models for Newtonian fluids that we can find for instance in [J.-F. Gerbeau, B. Perthame, Discrete Contin. Dyn. Syst. (2001)] which assume an appropriate relation between the Reynolds number and the aspect ratio with slip boundary condition at the bottom. Under a radial hypothesis at the leading order, for small Deborah number, we find an interesting formulation where polymeric effect changes the drag term in the second order shallow-water formulation (obtained by J.-F. Gerbeau, B. Perthame). We also discuss intermediate Deborah number with a fixed Reynolds number where a strong coupling is found through a nonlinear time-dependent Fokker–Planck equation. This generalizes, at a formal level, the derivation in [L. Chupin, Meth. Appl. Anal. (2009)] including non-linear effects (shallow-water framework).

How to cite

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Narbona-Reina, Gladys, and Bresch, Didier. "Two shallow-water type models for viscoelastic flows from kinetic theory for polymers solutions." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 47.6 (2013): 1627-1655. <http://eudml.org/doc/273148>.

@article{Narbona2013,
abstract = {In this work, depending on the relation between the Deborah, the Reynolds and the aspect ratio numbers, we formally derived shallow-water type systems starting from a micro-macro description for non-Newtonian fluids in a thin domain governed by an elastic dumbbell type model with a slip boundary condition at the bottom. The result has been announced by the authors in [G. Narbona-Reina, D. Bresch, Numer. Math. and Advanced Appl. Springer Verlag (2010)] and in the present paper, we provide a self-contained description, complete formal derivations and various numerical computations. In particular, we extend to FENE type systems the derivation of shallow-water models for Newtonian fluids that we can find for instance in [J.-F. Gerbeau, B. Perthame, Discrete Contin. Dyn. Syst. (2001)] which assume an appropriate relation between the Reynolds number and the aspect ratio with slip boundary condition at the bottom. Under a radial hypothesis at the leading order, for small Deborah number, we find an interesting formulation where polymeric effect changes the drag term in the second order shallow-water formulation (obtained by J.-F. Gerbeau, B. Perthame). We also discuss intermediate Deborah number with a fixed Reynolds number where a strong coupling is found through a nonlinear time-dependent Fokker–Planck equation. This generalizes, at a formal level, the derivation in [L. Chupin, Meth. Appl. Anal. (2009)] including non-linear effects (shallow-water framework).},
author = {Narbona-Reina, Gladys, Bresch, Didier},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {viscoelastic flows; polymers; Fokker–Planck equation; non newtonian fluids; Deborah number; shallow-water system; Fokker-Planck equation; non Newtonian fluids},
language = {eng},
number = {6},
pages = {1627-1655},
publisher = {EDP-Sciences},
title = {Two shallow-water type models for viscoelastic flows from kinetic theory for polymers solutions},
url = {http://eudml.org/doc/273148},
volume = {47},
year = {2013},
}

TY - JOUR
AU - Narbona-Reina, Gladys
AU - Bresch, Didier
TI - Two shallow-water type models for viscoelastic flows from kinetic theory for polymers solutions
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2013
PB - EDP-Sciences
VL - 47
IS - 6
SP - 1627
EP - 1655
AB - In this work, depending on the relation between the Deborah, the Reynolds and the aspect ratio numbers, we formally derived shallow-water type systems starting from a micro-macro description for non-Newtonian fluids in a thin domain governed by an elastic dumbbell type model with a slip boundary condition at the bottom. The result has been announced by the authors in [G. Narbona-Reina, D. Bresch, Numer. Math. and Advanced Appl. Springer Verlag (2010)] and in the present paper, we provide a self-contained description, complete formal derivations and various numerical computations. In particular, we extend to FENE type systems the derivation of shallow-water models for Newtonian fluids that we can find for instance in [J.-F. Gerbeau, B. Perthame, Discrete Contin. Dyn. Syst. (2001)] which assume an appropriate relation between the Reynolds number and the aspect ratio with slip boundary condition at the bottom. Under a radial hypothesis at the leading order, for small Deborah number, we find an interesting formulation where polymeric effect changes the drag term in the second order shallow-water formulation (obtained by J.-F. Gerbeau, B. Perthame). We also discuss intermediate Deborah number with a fixed Reynolds number where a strong coupling is found through a nonlinear time-dependent Fokker–Planck equation. This generalizes, at a formal level, the derivation in [L. Chupin, Meth. Appl. Anal. (2009)] including non-linear effects (shallow-water framework).
LA - eng
KW - viscoelastic flows; polymers; Fokker–Planck equation; non newtonian fluids; Deborah number; shallow-water system; Fokker-Planck equation; non Newtonian fluids
UR - http://eudml.org/doc/273148
ER -

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