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

Derivation and numerical approximation of hyperbolic viscoelastic flow systems: Saint-Venant 2D equations for Maxwell fluids