  We pursue here the development of models for complex (viscoelastic) fluids in
shallow free-surface gravity flows which was initiated by [Bouchut-Boyaval,
M3AS (23) 2013] for 1D (translation invariant) cases. The models we propose are
hyperbolic quasilinear systems that generalize Saint-Venant shallow-water
equations to incompressible Maxwell fluids. The models are compatible with a
formulation of the thermo-dynamics second principle. In comparison with
Saint-Venant standard shallow-water model, the momentum balance includes
extra-stresses associated with an elastic potential energy in addition to a
hydrostatic pressure. The extra-stresses are determined by an additional tensor
variable solution to a differential equation with various possible time rates.
For the numerical evaluation of solutions to Cauchy problems, we also propose
explicit schemes discretizing our generalized Saint-Venant systems with
Finite-Volume approximations that are entropy-consistent (under a CFL
constraint) in addition to satisfy exact (discrete) mass and momentum
conservation laws. In comparison with most standard viscoelastic numerical
models, our discrete models can be used for any retardation-time values (i.e.
in the vanishing "solvent-viscosity" limit). We finally illustrate our
hyperbolic viscoelastic flow models numerically using computer simulations in
benchmark test cases. On extending to Maxwell fluids some free-shear flow
testcases that are standard benchmarks for Newtonian fluids, we first show that
our (numerical) models reproduce well the viscoelastic physics,
phenomenologically at least, with zero retardation-time. Moreover, with a view
to quantitative evaluations, numerical results in the lid-driven cavity
testcase show that, in fact, our models can be compared with standard
viscoelastic flow models in sheared-flow benchmarks on adequately choosing the
physical parameters of our models. Analyzing our models asymptotics should
therefore shed new light on the famous High-Weissenberg Number Problem (HWNP),
which is a limit for all the existing viscoelastic numerical models.

Numerical Analysis;Classical Physics