We propose a unified framework to derive thin-layer reduced models for some
shallow free-surface flows driven by gravity. It applies to incompressible homogeneous fluids
whose momentum evolves according to Navier-Stokes equations,
with stress satisfying a rheology of viscous type (i.e. the standard Newtonian
law with a constant viscosity, but also non-Newtonian laws generalized to purely
viscous fluids and to viscoelastic fluids as well). For a given rheology, we derive
various thin-layer reduced models for flows on a rugous topography slowly varying
around an inclined plane. This is achieved thanks to a coherent simplification procedure,
which is formal but based on a mathematically clear consistency
requirement between scaling assumptions and the approximation errors in the
differential equations. The various thin-layer reduced models are obtained depending
on flow regime assumptions (either fast/inertial or slow/viscous). As far
as we know, it is the first time that the various thin-layer reduced models investigated here
are derived within the same mathematical framework. Furthermore,
we obtain new reduced models in the case of viscoelastic non-Newtonian fluids,
which extends [Bouchut & Boyaval, M3AS (23) 8, 2013].

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