Fourteen moment theory for granular gases

Kremer, Gilberto M.; Marques, Wilson

doi:10.3934/krm.2011.4.317

Cited by 26 publications

(70 citation statements)

References 9 publications

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“…Among themselves, m * ijk decays with time t * faster than any other moment, followed by R * ij , σ * ij , q * i and ∆ (in the order of fast to slow). That the stress (dashed red line) decays faster than the heat flux (dotted blue line) which decays faster than ∆/15 (magenta line with symbols) is in agreement with the findings of Kremer & Marques Jr. (2011).…”

Section: 2supporting

confidence: 87%

“…We further investigate the linear stability of HCS by scrutinizing the eigenmodes of longitudinal and transverse problems associated with the G26 system and other subsystems. Similar problems related to the stability of eigenmodes have been investigated earlier by Brey et al (1996) for five moment theory and by Kremer & Marques Jr. (2011) for thirteen moment theory. We find that one eigenmode from each Grad moment theory considered in this paper is unstable for inelastic gases while others are stable, while all the eigenmodes are stable for monatomic (elastic) gases.…”

Section: Introductionsupporting

confidence: 55%

“…We find that one eigenmode from each Grad moment theory considered in this paper is unstable for inelastic gases while others are stable, while all the eigenmodes are stable for monatomic (elastic) gases. Moreover, the thirteen moment theory of Kremer & Marques Jr. (2011) shows that the unstable heat mode of the longitudinal system remains unstable below a certain value of the coefficient of restitution for all wavenumbers while the present work reveals that the unstable heat mode of the longitudinal system associated with all moment theories expect for the G13 theory becomes stable above a critical wavenumber for all values of the coefficient of restitution. By investigating the critical wavenumbers of the longitudinal and transverse systems from the G26 theory, we also study the critical system size, beyond which the system becomes unstable, and compare the results with the existing theories and simulations.…”

Section: Introductionsupporting

confidence: 47%

“…It is commonly accepted that the dependence of heat flux on the density gradient in addition to the temperature gradient is akin to inclusion of the full trace of the fourth moment into the moment system (Kremer & Marques Jr. 2011;Garzó 2013). In that sense, the G13 and G20 theories for granular flows may not lead to meaningful results.…”

Section: Grad 26-moment Closurementioning

confidence: 99%

“…We assume a spatially homogeneous state (i.e., ∂(·) ∂xi = 0) without any external force acting on the particles of the granular gas (i.e., F = 0). Furthermore, following Kremer & Marques Jr. (2011), in the production terms (eqs. (A 1)-(A 6)), we shall retain only those nonlinear terms which are the product of the scalar fourth moment (w) and a vectorial or tensorial moment, and all other nonlinear terms are simply ignored.…”

Section: Homogeneous Cooling State Of a Freely Cooling Granular Gasmentioning

confidence: 99%

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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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Section: 2supporting

confidence: 87%

Section: Introductionsupporting

confidence: 55%

Section: Introductionsupporting

confidence: 47%

Section: Grad 26-moment Closurementioning

confidence: 99%

Section: Homogeneous Cooling State Of a Freely Cooling Granular Gasmentioning

confidence: 99%

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

2017

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Navier–Stokes Transport Coefficients for Monocomponent Granular Gases. I. Theoretical Results

Garzó

2019

Granular Gaseous Flows

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Modeling and analysis of time-dependent processes in a chemically reactive mixture

Ramos

Ribeiro

Soares

2017

Continuum Mech. Thermodyn.

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In this paper we study the propagation of sound waves and the dynamics of local wave disturbances induced by spontaneous internal fluctuations in a reactive mixture. We consider a non-diffusive, non-heat conducting and non-viscous mixture described by an Eulerian set of evolution equations. The model is derived from the kinetic theory in a hydrodynamic regime of a fast chemical reaction. The reactive source terms are explicitly computed from the kinetic theory and are built in the model in a proper way. For both time dependent problems, we first derive the appropriate dispersion relation, which retains the main effects of the chemical process, and then investigate the influence of the chemical reaction on the properties of interest in the problems studied here. We complete our study by developing a rather detailed analysis using the Hydrogen-Chlorine system as reference. Several numerical computations are included illustrating the behaviour of the phase velocity and attenuation coefficient in a low frequency regime and describing the spectrum of the eigenmodes in the small wavenumber limit.

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Fourteen moment theory for granular gases

Cited by 26 publications

References 9 publications

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

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

Navier–Stokes Transport Coefficients for Monocomponent Granular Gases. I. Theoretical Results

Modeling and analysis of time-dependent processes in a chemically reactive mixture

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