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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Shourick, Nathan 1, 2 ; Cheddadi, Ibrahim 2 ; Saramito, Pierre 1
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@article{CRMATH_2023__361_G11_1767_0,
author = {Shourick, Nathan and Cheddadi, Ibrahim and Saramito, Pierre},
title = {A new framework for shallow approximations of incompressible flows},
journal = {Comptes Rendus. Math\'ematique},
pages = {1767--1783},
year = {2023},
publisher = {Acad\'emie des sciences, Paris},
volume = {361},
number = {G11},
doi = {10.5802/crmath.526},
language = {en},
url = {https://www.numdam.org/articles/10.5802/crmath.526/}
}
TY - JOUR AU - Shourick, Nathan AU - Cheddadi, Ibrahim AU - Saramito, Pierre TI - A new framework for shallow approximations of incompressible flows JO - Comptes Rendus. Mathématique PY - 2023 SP - 1767 EP - 1783 VL - 361 IS - G11 PB - Académie des sciences, Paris UR - https://www.numdam.org/articles/10.5802/crmath.526/ DO - 10.5802/crmath.526 LA - en ID - CRMATH_2023__361_G11_1767_0 ER -
%0 Journal Article %A Shourick, Nathan %A Cheddadi, Ibrahim %A Saramito, Pierre %T A new framework for shallow approximations of incompressible flows %J Comptes Rendus. Mathématique %D 2023 %P 1767-1783 %V 361 %N G11 %I Académie des sciences, Paris %U https://www.numdam.org/articles/10.5802/crmath.526/ %R 10.5802/crmath.526 %G en %F CRMATH_2023__361_G11_1767_0
Shourick, Nathan; Cheddadi, Ibrahim; Saramito, Pierre. A new framework for shallow approximations of incompressible flows. Comptes Rendus. Mathématique, Tome 361 (2023) no. G11, pp. 1767-1783. doi: 10.5802/crmath.526
[1] A consistent thin-layer theory for Bingham plastics, J. Non-Newton. Fluid Mech., Volume 84 (1999) no. 1, pp. 65-81 | DOI | Zbl
[2] 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 | Zbl | MR
[3] Dynamics of polymeric liquids. Volume 2: Kinetic Theory, Wiley-Interscience, 1987
[4] A new model for shallow viscoelastic fluids, Math. Models Methods Appl. Sci., Volume 23 (2013) no. 8, pp. 1479-1526 | DOI | MR | Zbl
[5] 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] Gravity driven shallow water models for arbitrary topography, Commun. Math. Sci., Volume 2 (2004) no. 3, pp. 359-389 | DOI | MR | Zbl
[7] Shallow water equations for non-Newtonian fluids, J. Non-Newton. Fluid Mech., Volume 165 (2010) no. 13, pp. 712-732 | DOI | Zbl
[8] Derivation of Viscous Saint-Venant System for Laminar Shallow Water; Numerical Validation (2000) no. RR-4084 (Projet M3N) (Research Report)
[9] 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] 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] On the formulation of rheological equations of state, Proc. R. Soc. Lond., Ser. A, Volume 200 (1950), pp. 523-541 | Zbl | MR
[12] 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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