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

¹ Departamento de Matemática Aplicada I, E.T.S. Arquitectura, Universidad de Sevilla, Avda. Reina Mercedes 2, 41012 Sevilla, Spain.
gnarbona@us.es
² LAMA, UMR5127 CNRS, Université de Savoie, 73376 Le Bourget du Lac, France.
Didier.Bresch@univ-savoie.fr

Received: 8 November 2010
Revised: 21 September 2012

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).