Hyperbolic relaxation models for granular flows
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 44 (2010) no. 2, pp. 371-400.

In this work we describe an efficient model for the simulation of a two-phase flow made of a gas and a granular solid. The starting point is the two-velocity two-pressure model of Baer and Nunziato [Int. J. Multiph. Flow 16 (1986) 861-889]. The model is supplemented by a relaxation source term in order to take into account the pressure equilibrium between the two phases and the granular stress in the solid phase. We show that the relaxation process can be made thermodynamically coherent with an adequate choice of the granular stress. We then propose a numerical scheme based on a splitting approach. Each step of the time marching algorithm is made of two stages. In the first stage, the homogeneous convection equations are solved by a standard finite volume Rusanov scheme. In the second stage, the volume fraction is updated in order to take into account the equilibrium source term. The whole procedure is entropy dissipative. For simplified pressure laws (stiffened gas laws) we are able to prove that the approximated volume fraction stays within its natural bounds.

DOI : https://doi.org/10.1051/m2an/2010006
Classification : 76M12,  65M12
Mots clés : two-phase flow, hyperbolicity, relaxation, finite volume, entropy
     author = {Gallou\"et, Thierry and Helluy, Philippe and H\'erard, Jean-Marc and Nussbaum, Julien},
     title = {Hyperbolic relaxation models for granular flows},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     pages = {371--400},
     publisher = {EDP-Sciences},
     volume = {44},
     number = {2},
     year = {2010},
     doi = {10.1051/m2an/2010006},
     mrnumber = {2655954},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2010006/}
AU  - Gallouët, Thierry
AU  - Helluy, Philippe
AU  - Hérard, Jean-Marc
AU  - Nussbaum, Julien
TI  - Hyperbolic relaxation models for granular flows
JO  - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY  - 2010
DA  - 2010///
SP  - 371
EP  - 400
VL  - 44
IS  - 2
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an/2010006/
UR  - https://www.ams.org/mathscinet-getitem?mr=2655954
UR  - https://doi.org/10.1051/m2an/2010006
DO  - 10.1051/m2an/2010006
LA  - en
ID  - M2AN_2010__44_2_371_0
ER  - 
Gallouët, Thierry; Helluy, Philippe; Hérard, Jean-Marc; Nussbaum, Julien. Hyperbolic relaxation models for granular flows. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 44 (2010) no. 2, pp. 371-400. doi : 10.1051/m2an/2010006. http://www.numdam.org/articles/10.1051/m2an/2010006/

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