Abstract
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: \(\zeta \le \frac{1}{2}\) (\(\zeta =1\) is the corotational case and \(\zeta =0\) 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 \(G\rightarrow 0\).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
R.B. Bird, C.F. Curtiss, R.C. Armstrong, O. Hassager, Dynamics of Polymeric Liquids, Vol. 1: Fluid Mechanics (Wiley, New York, 1987)
F. Bouchut, Nonlinear Stability Of Finite Volume Methods for Hyperbolic Conservation Laws and Well-balanced Schemes for Sources, Frontiers in Mathematics (Birkhäuser Verlag, Basel, 2004). MR MR2128209 (2005m:65002)
F. Bouchut, S. Boyaval, A new model for shallow viscoelastic fluids. M3AS 23(8), 1479–1526 (2013)
F. Bouchut, S. Boyaval, Unified derivation of thin-layer reduced models for shallow free-surface gravity flows of viscous fluids. Eur. J. Mech. B/Fluids 55, Part 1, 116–131 (2016)
C.M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, vol. GM 325 (Springer, Berlin, 2000)
P.D. Lax, Hyperbolic systems of conservation laws ii. Commun. Pure Appl. Math. 10(4), 537–566 (1957)
P.G. LeFloch, Hyperbolic Systems of Conservation Laws: The Theory of Classical and Nonclassical Shock Waves, Lectures in Mathematics (ETH Zürich, Birkhäuser, Basel, 2002). MR 1927887 (2003j:35209)
M. Picasso, From the free surface flow of a viscoelastic fluid towards the elastic deformation of a solid. Comptes Rendus Mathmatique 1195(1), 1–92 (2016)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this paper
Cite this paper
Boyaval, S. (2018). Johnson–Segalman–Saint-Venant Equations for a 1D Viscoelastic Shallow Flow in Pure Elastic Limit. In: Klingenberg, C., Westdickenberg, M. (eds) Theory, Numerics and Applications of Hyperbolic Problems I. HYP 2016. Springer Proceedings in Mathematics & Statistics, vol 236. Springer, Cham. https://doi.org/10.1007/978-3-319-91545-6_16
Download citation
DOI: https://doi.org/10.1007/978-3-319-91545-6_16
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-91544-9
Online ISBN: 978-3-319-91545-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)