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).