Comptes Rendus
no. 5
Mathematical problems in mechanics
From the free surface flow of a viscoelastic fluid towards the elastic deformation of a solid

Presented by: the Editorial Board

Marco Picasso1
1 MATHICSE, Station 8, EPFL, 1015 Lausanne, Switzerland
Comptes Rendus. Mathématique, Volume 354 (2016) no. 5, pp. 543-548.

The free surface flow of an Oldroyd-B viscoelastic fluid is considered, following A. Bonito, M. Picasso, and M. Laso (2006) [1]. When removing a term in the extra-stress constitutive relation, the description of an elastic incompressible solid is obtained, in Eulerian coordinates, with the velocity field as the unknown, rather than the usual deformation field. Two simulations are proposed, a bouncing ball and an oscillating beam.

L'écoulement à frontière libre d'un fluide viscoélastique d'Oldroyd-B est considéré, selon A. Bonito, M. Picasso et M. Laso (2006) [1]. En supprimant un terme dans la loi constitutive de l'extra-contrainte, nous décrivons un solide élastique incompressible, en coordonnées eulériennes, avec le champ de vitesse comme inconnue, plutôt que le champ de déformation. Deux simulations sont proposées, le rebond d'une balle et une poutre vibrante.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2016.02.001

Marco Picasso 1

1 MATHICSE, Station 8, EPFL, 1015 Lausanne, Switzerland 
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Marco Picasso. From the free surface flow of a viscoelastic fluid towards the elastic deformation of a solid. Comptes Rendus. Mathématique, Volume 354 (2016) no. 5, pp. 543-548. doi : 10.1016/j.crma.2016.02.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.02.001/

[1] A. Bonito; M. Picasso; M. Laso Numerical simulation of 3D viscoelastic flows with free surfaces, J. Comput. Phys., Volume 215 (2006) no. 2, pp. 691-716

[2] A. Caboussat; M. Picasso; J. Rappaz Numerical simulation of free surface incompressible liquid flows surrounded by compressible gas, J. Comput. Phys., Volume 203 (2005) no. 2, pp. 626-649

[3] T. Dunne; R. Rannacher Adaptive finite element approximation of fluid–structure interaction based on an Eulerian variational formulation, Fluid–Structure Interaction, Lect. Notes Comput. Sci. Eng., vol. 53, Springer, Berlin, 2006, pp. 110-145

[4] R. Glowinski Finite Element Methods for Incompressible Viscous Flow, Handb. Numer. Anal., vol. IX, North-Holland, Amsterdam, 2003

[5] F.-H. Lin; C. Liu; P. Zhang On hydrodynamics of viscoelastic fluids, Commun. Pure Appl. Math., Volume 58 (2005) no. 11, pp. 1437-1471

[6] V. Maronnier; M. Picasso; J. Rappaz Numerical simulation of three-dimensional free surface flows, Int. J. Numer. Methods Fluids, Volume 42 (2003) no. 7, pp. 697-716

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