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

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.