Service interruption on Monday 11 July from 12:30 to 13:00: all the sites of the CCSD (HAL, EpiSciences, SciencesConf, AureHAL) will be inaccessible (network hardware connection).
Skip to Main content Skip to Navigation
Preprints, Working Papers, ...

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

Sébastien Boyaval 1, 2 
2 MATHERIALS - MATHematics for MatERIALS
CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique, Inria de Paris
Abstract : 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.
Complete list of metadata

Cited literature [42 references]  Display  Hide  Download

https://hal.archives-ouvertes.fr/hal-01661269
Contributor : Sébastien Boyaval Connect in order to contact the contributor
Submitted on : Tuesday, December 12, 2017 - 9:47:32 AM
Last modification on : Wednesday, June 8, 2022 - 12:50:04 PM

Files

hal.pdf
Files produced by the author(s)

Identifiers

  • HAL Id : hal-01661269, version 1
  • ARXIV : 1712.04257

Citation

Sébastien Boyaval. Derivation and numerical approximation of hyperbolic viscoelastic flow systems: Saint-Venant 2D equations for Maxwell fluids. 2017. ⟨hal-01661269⟩

Share

Metrics

Record views

286

Files downloads

199