  We propose a new reduced model for gravity-driven free-surface flows of
shallow elastic fluids. It is obtained by an asymptotic expansion of the
upper-convected Maxwell model for elastic fluids. The viscosity is assumed
small (of order epsilon, the aspect ratio of the thin layer of fluid), but the
relaxation time is kept finite. Additionally to the classical layer depth and
velocity in shallow models, our system describes also the evolution of two
scalar stresses. It has an intrinsic energy equation. The mathematical
properties of the model are established, an important feature being the
non-convexity of the physically relevant energy with respect to conservative
variables, but the convexity with respect to the physically relevant
pseudo-conservative variables. Numerical illustrations are given, based on a
suitable well-balanced finite-volume discretization involving an approximate
Riemann solver.
