A NEW MODEL FOR SHALLOW ELASTIC FLUIDS FRANÇOIS BOUCHUT AND SÉBASTIEN BOYAVAL Abstract. 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 con- servative 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. Contents 1. Introduction: thin layer approximations of non-Newtonian flows 2 2. Mathematical setting with the Upper-Convected Maxwell model for elastic fluids 3 3. Formal derivation of a thin layer approximation 5 4. The new reduced model and its mathematical properties 9 5. Finite volume method and numerical results 14 5.1. Approximate Riemann solver 14 5.2. Energy inequality 17 5.3. Numerical fluxes and CFL condition 20 5.4. Topography treatment 21 5.5. Numerical results 24 6. Conclusion 33 6.1. Physical interpretation from the macroscopic mechanical viewpoint 38 6.2. Physical interpretation at the microscopic molecular level 38 6.3. Open questions and perspectives 41 Appendix A. Convexity of the energy 41 References 42 Date : September 27, 2011. Key words and phrases. Viscoelastic fluids, Maxwell model, Oldroyd model, Saint Venant model, shallow-water, pseudo-conservative variables, well-balanced scheme. This work was completed while SB was visiting MATHICSE – ASN chair at EPFL. SB would like to thank Marco Picasso and Jacques Rappaz for their kind hospitality. 1