TY - GEN
T1 - Johnson–Segalman–Saint-venant equations for a 1d viscoelastic shallow flow in pure elastic limit
AU - Boyaval, Sébastien
N1 - Publisher Copyright:
© Springer International Publishing AG, part of Springer Nature 2018.
PY - 2018/1/1
Y1 - 2018/1/1
N2 - 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.
AB - 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.
KW - Generalized Saint-Venant equations
KW - Hyperbolic system
KW - Riemann problem
KW - Viscoelastic shallow flow
UR - https://www.scopus.com/pages/publications/85049379655
U2 - 10.1007/978-3-319-91545-6_16
DO - 10.1007/978-3-319-91545-6_16
M3 - Conference contribution
AN - SCOPUS:85049379655
SN - 9783319915449
T3 - Springer Proceedings in Mathematics and Statistics
SP - 205
EP - 213
BT - Theory, Numerics and Applications of Hyperbolic Problems I - Aachen, Germany, 2016
A2 - Westdickenberg, Michael
A2 - Klingenberg, Christian
PB - Springer New York LLC
T2 - 16th International Conference on Hyperbolic Problems: Theory, Numerics and Applications, 2016
Y2 - 1 August 2016 through 5 August 2016
ER -