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

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.