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

TL;DR

This paper extends the classical Saint-Venant shallow-water equations to include viscoelastic effects using the Johnson-Segalman model in the elastic limit, analyzing hyperbolicity, entropy, and solutions to the Riemann problem.

Contribution

It introduces a generalized 4x4 shallow-water model for viscoelastic fluids, proving hyperbolicity and entropy properties, and constructs solutions to the Riemann problem in the elastic limit.

Findings

The system is hyperbolic for small slip parameter $   1/2$.

The equations possess a natural mathematical entropy (free-energy).

Unique solutions to the Riemann problem are constructed under certain conditions.

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 $4 \times 4$ 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 $\zeta \le 1/2$ ($\zeta$ = 1 is the corotational case and $\zeta = 0$ the upper-convected Maxwell case). Moreover, it is naturally endowed with a mathematical entropy (a physical free-energy). When $\zeta \le 1/2$ 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 $G \to 0$ for small data.