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).
Keywords: viscoelastic flows, polymers, Fokker-Planck equation, non newtonian fluids, Deborah number, shallow-water system
@article{M2AN_2013__47_6_1627_0,
author = {Narbona-Reina, Gladys and Bresch, Didier},
title = {Two shallow-water type models for viscoelastic flows from kinetic theory for polymers solutions},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1627--1655},
year = {2013},
publisher = {EDP Sciences},
volume = {47},
number = {6},
doi = {10.1051/m2an/2013081},
mrnumber = {3110490},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2013081/}
}
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 PY - 2013 SP - 1627 EP - 1655 VL - 47 IS - 6 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2013081/ DO - 10.1051/m2an/2013081 LA - en ID - M2AN_2013__47_6_1627_0 ER -
%0 Journal Article %A Narbona-Reina, Gladys %A Bresch, Didier %T Two shallow-water type models for viscoelastic flows from kinetic theory for polymers solutions %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2013 %P 1627-1655 %V 47 %N 6 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/m2an/2013081/ %R 10.1051/m2an/2013081 %G en %F M2AN_2013__47_6_1627_0
Narbona-Reina, Gladys; Bresch, Didier. Two shallow-water type models for viscoelastic flows from kinetic theory for polymers solutions. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 47 (2013) no. 6, pp. 1627-1655. doi: 10.1051/m2an/2013081
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