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, Tome 47 (2013) no. 6, pp. 1627-1655.

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).

Classification : 76A05,  76A10,  35Q84,  82D60,  74D10,  35Q30,  78M35
Mots clés : viscoelastic flows, polymers, Fokker-Planck equation, non newtonian fluids, Deborah number, shallow-water system
     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 - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     pages = {1627--1655},
     publisher = {EDP-Sciences},
     volume = {47},
     number = {6},
     year = {2013},
     doi = {10.1051/m2an/2013081},
     mrnumber = {3110490},
     language = {en},
     url = {}
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 - Modélisation Mathématique et Analyse Numérique, Tome 47 (2013) no. 6, pp. 1627-1655. doi : 10.1051/m2an/2013081.

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