Comptes Rendus
Équations aux dérivées partielles, Mécanique
A new framework for shallow approximations of incompressible flows
Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1767-1783.

A new framework for the asymptotic analysis of incompressible flows of complex non-Newtonian materials is presented in this paper. It allows both to avoid redundant mathematical hypotheses and to dramatically reduce the amount of tedious formal calculations. The starting point of the proposed framework is a generic equation, easily adaptable to most problems of continuum mechanics, for which a thin-layer approximation is developed. We then show how to treat the so-called Gordon–Schowalter derivative, a general objective time derivative involved in non-Newtonian viscoelastic fluids. As a proof of concept of our framework, we apply it to the Maxwell viscoelastic model.

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DOI : 10.5802/crmath.526
Classification : 35Q35

Nathan Shourick 1, 2 ; Ibrahim Cheddadi 2 ; Pierre Saramito 1

1 Lab. Jean Kuntzmann – CNRS and Université Grenoble-Alpes, F-38041 Grenoble, France
2 Univ. Grenoble Alpes, CNRS, UMR 5525, VetAgro Sup, Grenoble INP, TIMC, 38000 Grenoble, France
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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     title = {A new framework for shallow approximations of incompressible flows},
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Nathan Shourick; Ibrahim Cheddadi; Pierre Saramito. A new framework for shallow approximations of incompressible flows. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1767-1783. doi : 10.5802/crmath.526. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.526/

[1] Neil J. Balmforth; Richard V. Craster A consistent thin-layer theory for Bingham plastics, J. Non-Newton. Fluid Mech., Volume 84 (1999) no. 1, pp. 65-81 | DOI | Zbl

[2] Noé Bernabeu; Pierre Saramito; Claude Smutek Numerical modeling of non-Newtonian viscoplastic flows. II: Viscoplastic fluids and general tridimensional topographies, Int. J. Numer. Anal. Model., Volume 11 (2014) no. 1, pp. 213-228 | MR | Zbl

[3] R. Byron Bird; Charles F. Curtiss; Robert C. Armstrong; Ole Hassager Dynamics of polymeric liquids. Volume 2: Kinetic Theory, Wiley-Interscience, 1987

[4] François Bouchut; Sébastien Boyaval A new model for shallow viscoelastic fluids, Math. Models Methods Appl. Sci., Volume 23 (2013) no. 8, pp. 1479-1526 | DOI | MR | Zbl

[5] François Bouchut; Sébastien Boyaval Unified derivation of thin-layer reduced models for shallow free-surface gravity flows of viscous fluids, Eur. J. Mech. B Fluids, Volume 55 (2016), pp. 116-131 | DOI | MR | Zbl

[6] François Bouchut; Michael Westdickenberg Gravity driven shallow water models for arbitrary topography, Commun. Math. Sci., Volume 2 (2004) no. 3, pp. 359-389 | DOI | MR | Zbl

[7] Enrique D. Fernández-Nieto; Pascal Noble; Jean-Paul Vila Shallow water equations for non-Newtonian fluids, J. Non-Newton. Fluid Mech., Volume 165 (2010) no. 13, pp. 712-732 | DOI | Zbl

[8] Jean-Frédéric Gerbeau; Benoît Perthame Derivation of Viscous Saint-Venant System for Laminar Shallow Water; Numerical Validation (2000) no. RR-4084 (Projet M3N) (Research Report)

[9] K. F. Liu; C. C. Mei Approximate equations for the slow spreading of a thin sheet of Bingham plastic fluid, Phys. Fluids, A, Volume 2 (1990) no. 1, pp. 30-36 | DOI | Zbl

[10] Fabien Marche Derivation of a new two-dimensional viscous shallow water model with varying topography, bottom friction and capillary effects, Eur. J. Mech. B Fluids, Volume 26 (2007) no. 1, pp. 49-63 | DOI | MR | Zbl

[11] James G. Oldroyd On the formulation of rheological equations of state, Proc. R. Soc. Lond., Ser. A, Volume 200 (1950), pp. 523-541 | MR | Zbl

[12] AJC Barré de Saint-Venant Théorie et équations générales du mouvement non permanent des eaux courantes, C. R. Acad. Sci., Paris, Volume 17 (1871) no. 73, pp. 147-154

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