Hydrodynamics of Inelastic Maxwell Models

V. Garzó; A. Santos

Mathematical Modelling of Natural Phenomena (2011)

  • Volume: 6, Issue: 4, page 37-76
  • ISSN: 0973-5348

Abstract

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An overview of recent results pertaining to the hydrodynamic description (both Newtonian and non-Newtonian) of granular gases described by the Boltzmann equation for inelastic Maxwell models is presented. The use of this mathematical model allows us to get exact results for different problems. First, the Navier–Stokes constitutive equations with explicit expressions for the corresponding transport coefficients are derived by applying the Chapman–Enskog method to inelastic gases. Second, the non-Newtonian rheological properties in the uniform shear flow (USF) are obtained in the steady state as well as in the transient unsteady regime. Next, an exact solution for a special class of Couette flows characterized by a uniform heat flux is worked out. This solution shares the same rheological properties as the USF and, additionally, two generalized transport coefficients associated with the heat flux vector can be identified. Finally, the problem of small spatial perturbations of the USF is analyzed with a Chapman–Enskog-like method and generalized (tensorial) transport coefficients are obtained.

How to cite

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Garzó, V., and Santos, A.. "Hydrodynamics of Inelastic Maxwell Models." Mathematical Modelling of Natural Phenomena 6.4 (2011): 37-76. <http://eudml.org/doc/222235>.

@article{Garzó2011,
abstract = {An overview of recent results pertaining to the hydrodynamic description (both Newtonian and non-Newtonian) of granular gases described by the Boltzmann equation for inelastic Maxwell models is presented. The use of this mathematical model allows us to get exact results for different problems. First, the Navier–Stokes constitutive equations with explicit expressions for the corresponding transport coefficients are derived by applying the Chapman–Enskog method to inelastic gases. Second, the non-Newtonian rheological properties in the uniform shear flow (USF) are obtained in the steady state as well as in the transient unsteady regime. Next, an exact solution for a special class of Couette flows characterized by a uniform heat flux is worked out. This solution shares the same rheological properties as the USF and, additionally, two generalized transport coefficients associated with the heat flux vector can be identified. Finally, the problem of small spatial perturbations of the USF is analyzed with a Chapman–Enskog-like method and generalized (tensorial) transport coefficients are obtained. },
author = {Garzó, V., Santos, A.},
journal = {Mathematical Modelling of Natural Phenomena},
keywords = {kinetic theory; Boltzmann equation; granular gases; inelastic Maxwell models; transport coefficients; hydrodynamics},
language = {eng},
month = {7},
number = {4},
pages = {37-76},
publisher = {EDP Sciences},
title = {Hydrodynamics of Inelastic Maxwell Models},
url = {http://eudml.org/doc/222235},
volume = {6},
year = {2011},
}

TY - JOUR
AU - Garzó, V.
AU - Santos, A.
TI - Hydrodynamics of Inelastic Maxwell Models
JO - Mathematical Modelling of Natural Phenomena
DA - 2011/7//
PB - EDP Sciences
VL - 6
IS - 4
SP - 37
EP - 76
AB - An overview of recent results pertaining to the hydrodynamic description (both Newtonian and non-Newtonian) of granular gases described by the Boltzmann equation for inelastic Maxwell models is presented. The use of this mathematical model allows us to get exact results for different problems. First, the Navier–Stokes constitutive equations with explicit expressions for the corresponding transport coefficients are derived by applying the Chapman–Enskog method to inelastic gases. Second, the non-Newtonian rheological properties in the uniform shear flow (USF) are obtained in the steady state as well as in the transient unsteady regime. Next, an exact solution for a special class of Couette flows characterized by a uniform heat flux is worked out. This solution shares the same rheological properties as the USF and, additionally, two generalized transport coefficients associated with the heat flux vector can be identified. Finally, the problem of small spatial perturbations of the USF is analyzed with a Chapman–Enskog-like method and generalized (tensorial) transport coefficients are obtained.
LA - eng
KW - kinetic theory; Boltzmann equation; granular gases; inelastic Maxwell models; transport coefficients; hydrodynamics
UR - http://eudml.org/doc/222235
ER -

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