High-Degree Collisional Moments of Inelastic Maxwell Mixtures—Application to the Homogeneous Cooling and Uniform Shear Flow States
The Boltzmann equation for <i>d</i>-dimensional inelastic Maxwell models is considered to determine the collisional moments of the second, third and fourth degree in a granular binary mixture. These collisional moments are exactly evaluated in terms of the velocity moments of the distrib...
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MDPI AG
2023-01-01
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| Online Access: | https://www.mdpi.com/1099-4300/25/2/222 |
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| author | Constantino Sánchez Romero Vicente Garzó |
| author_facet | Constantino Sánchez Romero Vicente Garzó |
| author_sort | Constantino Sánchez Romero |
| collection | DOAJ |
| description | The Boltzmann equation for <i>d</i>-dimensional inelastic Maxwell models is considered to determine the collisional moments of the second, third and fourth degree in a granular binary mixture. These collisional moments are exactly evaluated in terms of the velocity moments of the distribution function of each species when diffusion is absent (mass flux of each species vanishes). The corresponding associated eigenvalues as well as cross coefficients are obtained as functions of the coefficients of normal restitution and the parameters of the mixture (masses, diameters and composition). The results are applied to the analysis of the time evolution of the moments (scaled with a thermal speed) in two different nonequilibrium situations: the homogeneous cooling state (HCS) and the uniform (or simple) shear flow (USF) state. In the case of the HCS, in contrast to what happens for simple granular gases, it is demonstrated that the third and fourth degree moments could diverge in time for given values of the parameters of the system. An exhaustive study on the influence of the parameter space of the mixture on the time behavior of these moments is carried out. Then, the time evolution of the second- and third-degree velocity moments in the USF is studied in the tracer limit (namely, when the concentration of one of the species is negligible). As expected, while the second-degree moments are always convergent, the third-degree moments of the tracer species can be also divergent in the long time limit. |
| first_indexed | 2024-03-11T08:51:41Z |
| format | Article |
| id | doaj.art-451831ad957b4324a99a02d4209e6189 |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-03-11T08:51:41Z |
| publishDate | 2023-01-01 |
| publisher | MDPI AG |
| record_format | Article |
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| spelling | doaj.art-451831ad957b4324a99a02d4209e61892023-11-16T20:22:30ZengMDPI AGEntropy1099-43002023-01-0125222210.3390/e25020222High-Degree Collisional Moments of Inelastic Maxwell Mixtures—Application to the Homogeneous Cooling and Uniform Shear Flow StatesConstantino Sánchez Romero0Vicente Garzó1Departamento de Física, Universidad de Extremadura, Avda. de Elvas s/n, E-06006 Badajoz, SpainDepartamento de Física and Instituto de Computación Científica Avanzada (ICCAEx), Universidad de Extremadura, Avda. de Elvas s/n, E-06006 Badajoz, SpainThe Boltzmann equation for <i>d</i>-dimensional inelastic Maxwell models is considered to determine the collisional moments of the second, third and fourth degree in a granular binary mixture. These collisional moments are exactly evaluated in terms of the velocity moments of the distribution function of each species when diffusion is absent (mass flux of each species vanishes). The corresponding associated eigenvalues as well as cross coefficients are obtained as functions of the coefficients of normal restitution and the parameters of the mixture (masses, diameters and composition). The results are applied to the analysis of the time evolution of the moments (scaled with a thermal speed) in two different nonequilibrium situations: the homogeneous cooling state (HCS) and the uniform (or simple) shear flow (USF) state. In the case of the HCS, in contrast to what happens for simple granular gases, it is demonstrated that the third and fourth degree moments could diverge in time for given values of the parameters of the system. An exhaustive study on the influence of the parameter space of the mixture on the time behavior of these moments is carried out. Then, the time evolution of the second- and third-degree velocity moments in the USF is studied in the tracer limit (namely, when the concentration of one of the species is negligible). As expected, while the second-degree moments are always convergent, the third-degree moments of the tracer species can be also divergent in the long time limit.https://www.mdpi.com/1099-4300/25/2/222Boltzmann equationgranular mixturesinelastic Maxwell modelscollisional momentshomogeneous cooling stateuniform shear flow state |
| spellingShingle | Constantino Sánchez Romero Vicente Garzó High-Degree Collisional Moments of Inelastic Maxwell Mixtures—Application to the Homogeneous Cooling and Uniform Shear Flow States Entropy Boltzmann equation granular mixtures inelastic Maxwell models collisional moments homogeneous cooling state uniform shear flow state |
| title | High-Degree Collisional Moments of Inelastic Maxwell Mixtures—Application to the Homogeneous Cooling and Uniform Shear Flow States |
| title_full | High-Degree Collisional Moments of Inelastic Maxwell Mixtures—Application to the Homogeneous Cooling and Uniform Shear Flow States |
| title_fullStr | High-Degree Collisional Moments of Inelastic Maxwell Mixtures—Application to the Homogeneous Cooling and Uniform Shear Flow States |
| title_full_unstemmed | High-Degree Collisional Moments of Inelastic Maxwell Mixtures—Application to the Homogeneous Cooling and Uniform Shear Flow States |
| title_short | High-Degree Collisional Moments of Inelastic Maxwell Mixtures—Application to the Homogeneous Cooling and Uniform Shear Flow States |
| title_sort | high degree collisional moments of inelastic maxwell mixtures application to the homogeneous cooling and uniform shear flow states |
| topic | Boltzmann equation granular mixtures inelastic Maxwell models collisional moments homogeneous cooling state uniform shear flow state |
| url | https://www.mdpi.com/1099-4300/25/2/222 |
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