An Introduction to the Boltzmann Equation and Transport Processes in Gases

Kremer, Gilberto M.

doi:10.1007/978-3-642-11696-4

Cited by 220 publications

(262 citation statements)

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“…This defines a conservative system (i.e., the total kinetic energy is conserved by collisions) that has been used for a long time to model polyatomic gases [6,48,49,55]. The results obtained in this limit are displayed in the last col- Table II and fully agree with those first derived by Pidduck [48].…”

Section: A Limiting Casessupporting

confidence: 76%

“…As a consequence, E = 0 and one recovers the known results for inelastic smooth particles [11]. Thus, the presence of roughness induces a non-vanishing function E, even in the conservative case of perfectly rough particles (α = β = 1) [6,55]. A subtler consequence of roughness is the symmetry breakdown of the traceless tensor C ij .…”

Section: B First-order Distributionsupporting

confidence: 76%

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Transport coefficients of a granular gas of inelastic rough hard spheres

Kremer¹,

Santos

Garzó

2014

Phys. Rev. E

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The Boltzmann equation for inelastic and rough hard spheres is considered as a model of a dilute granular gas. In this model, the collisions are characterized by constant coefficients of normal and tangential restitution and hence the translational and rotational degrees of freedom are coupled. A normal solution to the Boltzmann equation is obtained by means of the Chapman-Enskog method for states near the homogeneous cooling state. The analysis is carried out to first order in the spatial gradients of the number density, the flow velocity, and the granular temperature. The constitutive equations for the momentum and heat fluxes and for the cooling rate are derived, and the associated transport coefficients are expressed in terms of the solutions of linear integral equations. For practical purposes, a first Sonine approximation is used to obtain explicit expressions of the transport coefficients as nonlinear functions of both coefficients of restitution and the moment of inertia. Known results for purely smooth inelastic spheres and perfectly elastic and rough spheres are recovered in the appropriate limits.

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Section: A Limiting Casessupporting

confidence: 76%

Section: B First-order Distributionsupporting

confidence: 76%

Transport coefficients of a granular gas of inelastic rough hard spheres

Kremer¹,

Santos

Garzó

2014

Phys. Rev. E

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“…where i, j = 1, 2, according to the particle species [36,38]. Therefore, the relative velocity between colliding particles would be w i j ≡ w i − w j .…”

Section: Theory and Simulation Outlinementioning

confidence: 99%

“…1. A rough hardsphere collisional model with two velocity-independent coefficients (tangential, β, and normal, α, restitution coefficients) seems to describe more accurately a generic collision between two dry grains [36].…”

Section: Introductionmentioning

confidence: 97%

Thermal properties of an impurity immersed in a granular gas of rough hard spheres

et al. 2017

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Abstract. We study in this work the dynamics of a granular impurity immersed in a low-density granular gas of identical particles. For description of the kinetics of the granular gas and the impurity particles we use the rough hard sphere collisional model. We take into account the effects of non-conservation of energy upon particle collision. We find an (approximate) analytical solution of the pertinent kinetic equations for the singleparticle velocity distribution functions that reproduces reasonably well the properties of translational/rotational energy non-equipartition. We assess the accuracy of the theoretical solution by comparing with computer simulations. For this, we use two independent computer data sets, from molecular dynamics (MD) and from Direct Simulation Monte Carlo method (DSMC). Both approach well, with different degrees, the kinetic theory within a reasonable range of parameter values.

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“…In order to obtain a closed (finite) system of moment equations, Grad (1949b) expanded the velocity distribution function f in a finite linear combination of the N -dimensional Hermite polynomials (Grad 1949a) in peculiar velocity and computed the unknown coefficients in the expansion in terms of the considered moments by satisfying their definitions with the approximated distribution function. This method of obtaining a closed set of moment equations is referred to as Grad's method of moments and its details can be found in Grad (1949b) and in many standard textbooks, see e.g., Struchtrup (2005); Kremer (2010).…”

Section: Grad's Methods Of Momentsmentioning

confidence: 99%

Higher-order moment theories for dilute granular gases of smooth hard spheres

2017

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Grad's method of moments is employed to develop higher-order Grad moment equations-up to first 26-moments-for dilute granular gases within the framework of the (inelastic) Boltzmann equation. The homogeneous cooling state of a freely cooling granular gas is investigated with the Grad 26-moment equations in a semi-linearized setting and it is shown that the granular temperature in the homogeneous cooling state still decays according to Haff's law while the other higher-order moments decay on a faster time scale. The nonlinear terms of fully contracted fourth moment are also considered and, by exploiting the stability analysis of fixed points, it is shown that these nonlinear terms have negligible effect on Haff's law. Furthermore, an even larger Grad moment system which includes the fully contracted sixth moment is also scrutinized and the stability analysis of fixed points is again exploited to conclude that even the inclusion of scalar sixth order moment into the Grad moment system has negligible effect on Haff's law. The constitutive relations for the stress and heat flux (i.e., the Navier-Stokes and Fourier relations) are derived through the Grad 26-moment equations and compared with those obtained via CE expansion and via computer simulations. The linear stability of the homogeneous cooling state is analyzed through the Grad 26-moment system and various sub-systems by decomposing them into longitudinal and transverse systems. It is found that one eigenmode in both longitudinal and transverse systems in the case of inelastic gases is unstable. By comparing the eigenmodes from various theories, it is established that the 13-moment eigenmode theory predicts that the unstable heat mode of the longitudinal system remains unstable for all wavenumbers below a certain coefficient of restitution while any other higher-order moment theory shows that this mode becomes stable above some critical wavenumber for all values of the coefficient of restitution. In particular, the Grad 26-moment theory leads to a smooth profile for the critical wavenumber in contrast to the other considered theories. Furthermore, the critical system size obtained through the Grad 26-moment and existing theories are also in excellent agreement.

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An Introduction to the Boltzmann Equation and Transport Processes in Gases

Cited by 220 publications

References 0 publications

Transport coefficients of a granular gas of inelastic rough hard spheres

Transport coefficients of a granular gas of inelastic rough hard spheres

Thermal properties of an impurity immersed in a granular gas of rough hard spheres

Higher-order moment theories for dilute granular gases of smooth hard spheres

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