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
Numerical illustrations are given, based on a suitable well-balanced finite-volume
discretization involving an approximate Riemann solver.
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