A new model for shallow viscoelastic fluids

François Bouchut (1) , Sébastien Boyaval (2, 3)

1 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
2 LHSV - Laboratoire d'Hydraulique Saint-Venant / Saint-Venant Laboratory for Hydraulics
3 MICMAC - Methods and engineering of multiscale computing from atom to continuum

We propose a new reduced model for gravity-driven free-surface flows of shallow elastic fluids. It is obtained by an asymptotic expansion of the upper-convected Maxwell model for elastic fluids. The viscosity is assumed small (of order epsilon, the aspect ratio of the thin layer of fluid), but the relaxation time is kept finite. Additionally to the classical layer depth and velocity in shallow models, our system describes also the evolution of two scalar stresses. It has an intrinsic energy equation. The mathematical properties of the model are established, an important feature being the non-convexity of the physically relevant energy with respect to conservative variables, but the convexity with respect to the physically relevant pseudo-conservative variables. Numerical illustrations are given, based on a suitable well-balanced finite-volume discretization involving an approximate Riemann solver.

Mots clés: Viscoelastic fluids; Maxwell model; Oldroyd model; Saint Venant equations; shallow-water model; pseudo-conservative hyperbolic system of equations; well-balanced finite-volume scheme

Domaines: Analyse numérique [math.NA]; Mécanique des fluides [physics.class-ph]; Mécanique des fluides [physics.class-ph]; Equations aux dérivées partielles [math.AP]

François Bouchut, Sébastien Boyaval. A new model for shallow viscoelastic fluids. Mathematical Models and Methods in Applied Sciences, 2013, 23 (8), pp.1479-1526. ⟨10.1142/S0218202513500140⟩. ⟨hal-00628651v2⟩