A new framework for shallow approximations of incompressible flows

Nathan Shourick; Ibrahim Cheddadi; Pierre Saramito

doi:10.5802/crmath.526

Équations aux dérivées partielles, Mécanique

A new framework for shallow approximations of incompressible flows

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

Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1767-1783.

Résumé

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.

Reçu le : 2023-04-07
Révisé le : 2023-06-14
Accepté le : 2023-06-14
Publié le : 2023-12-21

DOI : 10.5802/crmath.526

Classification : 35Q35

Affiliations des auteurs :

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

@article{CRMATH_2023__361_G11_1767_0,
     author = {Nathan Shourick and Ibrahim Cheddadi and Pierre Saramito},
     title = {A new framework for shallow approximations of incompressible flows},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1767--1783},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {361},
     year = {2023},
     doi = {10.5802/crmath.526},
     language = {en},
}

TY  - JOUR
AU  - Nathan Shourick
AU  - Ibrahim Cheddadi
AU  - Pierre Saramito
TI  - A new framework for shallow approximations of incompressible flows
JO  - Comptes Rendus. Mathématique
PY  - 2023
SP  - 1767
EP  - 1783
VL  - 361
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.526
LA  - en
ID  - CRMATH_2023__361_G11_1767_0
ER  -

%0 Journal Article
%A Nathan Shourick
%A Ibrahim Cheddadi
%A Pierre Saramito
%T A new framework for shallow approximations of incompressible flows
%J Comptes Rendus. Mathématique
%D 2023
%P 1767-1783
%V 361
%I Académie des sciences, Paris
%R 10.5802/crmath.526
%G en
%F CRMATH_2023__361_G11_1767_0

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/

Bibliographie
Cité par

[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

Cité par Sources :

Commentaires - Politique