A new model for shallow viscoelastic fluids

TL;DR

This paper introduces a novel reduced model for shallow elastic fluids that captures elastic stresses and energy dynamics, derived from the upper-convected Maxwell model, with mathematical analysis and numerical validation.

Contribution

It presents a new asymptotic shallow model for elastic fluids including stress evolution, energy properties, and numerical methods, extending classical shallow flow models.

Findings

The model accurately describes elastic stresses in shallow flows.

The energy function is non-convex in conservative variables but convex in pseudo-conservative variables.

Numerical simulations demonstrate the model's effectiveness.

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 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.