Johnson–Segalman–Saint-venant equations for a 1d viscoelastic shallow flow in pure elastic limit

The shallow-water equations of Saint-Venant, often used to model the long-wave dynamics of free-surface gravity flows governed by inertia and hydrostatic pressure, can be generalized to account for the elongational rheology of non-Newtonian fluids too. We consider here 1D shallow-water equations generalized to viscoelastic fluids using the Johnson–Segalman model in pure elastic limit (i.e., at infinitely-large Deborah number, when source terms vanish). The quasilinear system of first-order equations is hyperbolic when the slip parameter is small: β ≤ 1/2 (β =1 is the corotational case and β =0 the upper-convected Maxwell case). It is naturally endowed with a mathematical entropy (a physical free-energy), and it is strictly hyperbolic when vacuum is excluded. Then, for any initial data, we construct the unique solution to the Riemann problem under Lax admissibility conditions. The standard Saint-Venant case is recovered for small data in the non-elastic limit G → 0.