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Johnson-Segalman – Saint-Venant equations for viscoelastic shallow flows in the elastic limit

Sébastien Boyaval 1, 2
CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique, Inria de Paris
Abstract : The shallow-water equations of Saint-Venant, often used to model the long-wave dynamics of free-surface flows driven by inertia and hydrostatic pressure, can be generalized to account for the elongational rheology of non-Newtonian fluids too. We consider here the 4 × 4 shallow-water equations generalized to viscoelastic fluids using the Johnson-Segalman model in the elastic limit (i.e. at infinitely-large Deborah number, when source terms vanish). The system of nonlinear first-order equations is hyperbolic when the slip parameter is small ζ ≤ 1/2 (ζ = 1 is the corotational case and ζ = 0 the upper-convected Maxwell case). Moreover, it is naturally endowed with a mathematical entropy (a physical free-energy). When ζ ≤ 1/2 and for any initial data excluding vacuum, we construct here, when elasticity G > 0 is non-zero, the unique solution to the Riemann problem under Lax admissibility conditions. The standard Saint-Venant case is recovered when G → 0 for small data.
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Contributor : Sébastien Boyaval <>
Submitted on : Thursday, November 24, 2016 - 10:30:23 PM
Last modification on : Monday, December 14, 2020 - 5:25:31 PM


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  • HAL Id : hal-01402628, version 1
  • ARXIV : 1611.08491


Sébastien Boyaval. Johnson-Segalman – Saint-Venant equations for viscoelastic shallow flows in the elastic limit. XVI International Conference on Hyperbolic Problems Theory, Numerics, Applications (Hyp2016), Aug 2016, Aachen, Germany. ⟨hal-01402628⟩



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