Séance Séminaire

Séminaire ACSIOM

Thin-layer modelling of shallow viscoelastic free-surface flows

We will present a reduced model for gravity-driven free-surface flows of shallow viscoelastic fluids (work in collaboration with F. Bouchut). It is obtained by a formal asymptotic expansion of the upper-convected Maxwell model when the viscosity is small like the aspect ratio of the thin layer of fluid, while the relaxation time is kept finite. Additionally to the classical layer depth and velocity in shallow water models, our system describes also the evolution of two scalar stress variables. We will discuss some mathematical properties of the model. (i) It is a hyperbolic system of first-order PDEs close to Saint-Venant equations, with two additional equations (for stresses) and an intrinsic energy equation. (ii) The physically relevant energy is not convex with respect to the conservative variables, but with respect to other (more physically relevant) "pseudo"-conservative variables. We will describe a suitable finite-volume discretization in the "pseudo"-conservative variables involving an approximate Riemann solver. Numerical simulations will finally illustrate the reduced model.