On the first Sonine correction for granular gases

Coppex, François; Droz, Michel; Piasecki, J.; Trizac, Emmanuel

doi:10.1016/s0378-4371(03)00593-4

Cited by 18 publications

(29 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The solution of the resulting linear equation for a 2 had the structure of a polynomial of fourth degree in α 2 divided by a polynomial of eighth degree in α (with no α 5 and α 7 terms). Although promising, this alternative method yields poor results for small and moderate inelasticities and only improves over the vNE benchmark formula if α 0.4, as comparison with DSMC data for d = 2 shows [16]. Coppex et al also elaborated further on the ambiguity of the linear approximation for a 2 pointed out in Ref.…”

Section: A Brief Review Of Previous Resultsmentioning

confidence: 99%

See 1 more Smart Citation

The second and third Sonine coefficients of a freely cooling granular gas revisited

Santos

Montanero

2009

Granular Matter

View full text Add to dashboard Cite

In its simplest statistical-mechanical description, a granular fluid can be modeled as composed of smooth inelastic hard spheres (with a constant coefficient of normal restitution α) whose velocity distribution function obeys the Enskog-Boltzmann equation. The basic state of a granular fluid is the homogeneous cooling state, characterized by a homogeneous, isotropic, and stationary distribution of scaled velocities, F(c). The behavior of F(c) in the domain of thermal velocities (c ∼ 1) can be characterized by the two first non-trivial coefficients (a 2 and a 3 ) of an expansion in Sonine polynomials. The main goals of this paper are to review some of the previous efforts made to estimate (and measure in computer simulations) the α-dependence of a 2 and a 3 , to report new computer simulations results of a 2 and a 3 for two-dimensional systems, and to investigate the possibility of proposing theoretical estimates of a 2 and a 3 with an optimal compromise between simplicity and accuracy.

show abstract

Section: A Brief Review Of Previous Resultsmentioning

confidence: 99%

“…Other class-I approximations for a 2 were considered by Coppex et al [16] and are more generally described in Appendix B.…”

Section: Theoretical Estimates From Linear Approximationsmentioning

confidence: 99%

The second and third Sonine coefficients of a freely cooling granular gas revisited

Santos

Montanero

2009

Granular Matter

View full text Add to dashboard Cite

show abstract

“…This is what happens for α > ∼ 0.7. However, for larger inelasticity, the fourth cumulant a 2 is not negligible [25,26,29,30]. Since a 2 > 0 for α < ∼ 0.7, then the standard estimates for the collision frequencies are smaller than their modified counterparts.…”

Section: Modified First Sonine Approximationmentioning

confidence: 95%

Modified Sonine approximation for the Navier–Stokes transport coefficients of a granular gas

Garzó

Santos

Montanero

2007

Physica A: Statistical Mechanics and its Applications

144

View full text Add to dashboard Cite

Motivated by the disagreement found at high dissipation between simulation data for the heat flux transport coefficients and the expressions derived from the Boltzmann equation by the standard first Sonine approximation [Brey et al., Phys. Rev. E 70, 051301 (2004); J. Phys.: Condens. Matter 17, S2489 (2005)], we implement in this paper a modified version of the first Sonine approximation in which the Maxwell-Boltzmann weight function is replaced by the homogeneous cooling state distribution. The structure of the transport coefficients is common in both approximations, the distinction appearing in the coefficient of the fourth cumulant a2. Comparison with computer simulations shows that the modified approximation significantly improves the estimates for the heat flux transport coefficients at strong dissipation. In addition, the slight discrepancies between simulation and the standard first Sonine estimates for the shear viscosity and the self-diffusion coefficient are also partially corrected by the modified approximation. Finally, the extension of the modified first Sonine approximation to the transport coefficients of the Enskog kinetic theory is presented.

show abstract

“…A first approximation for the velocity pdf is therefore to truncate the expansion at second order (p = 2). An approximated expression for the coefficient a 2 has been found as a function of the restitution coefficient α and the dimension d [22,23,24]. It must be noted that this approximation is only valid for not too large velocities, since the tails of the pdf have been shown [22] to be overpopulated with respect to the Gaussian distribution.…”

Section: B the Velocity Distribution Functionmentioning

confidence: 99%

Fluctuations of Power Injection in Randomly Driven Granular Gases

et al. 2006

Self Cite

View full text Add to dashboard Cite

We investigate the large deviation function π∞(w) for the fluctuations of the power W(t) = wt, integrated over a time t, injected by a homogeneous random driving into a granular gas, in the infinite time limit. Our analytical study starts from a generalized Liouville equation and exploits a Molecular Chaos-like assumption. We obtain an equation for the generating function of the cumulants μ(λ) which appears as a generalization of the inelastic Boltzmann equation and has a clear physical interpretation. Reasonable assumptions are used to obtain μ(λ) in a closed analytical form. A Legendre transform is sufficient to get the large deviation function π∞(w). Our main result, apart from an estimate of all the cumulants of W(t) at large times t, is that π∞ has no negative branch. This immediately results in the inapplicability of the Gallavotti-Cohen Fluctuation Relation (GCFR), that in previous studies had been suggested to be valid for injected power in driven granular gases. We also present numerical results, in order to discuss the finite time behavior of the fluctuations of W (t) . We discover that their probability density function converges extremely slowly to its asymptotic scaling form: the third cumulant saturates after a characteristic time τ larger than ∼50 mean free times and the higher order cumulants evolve even slower. The asymptotic value is in good agreement with our theory. Remarkably, a numerical check of the GCFR is feasible only at small times (at most τ/10), since negative events disappear at larger times. At such small times this check leads to the misleading conclusion that GCFR is satisfied for π∞(w). We offer an explanation for this remarkable apparent verification. In the inelastic Maxwell model, where a better statistics can be achieved, we are able to numerically observe the “failure” of GCFR

show abstract

On the first Sonine correction for granular gases

Cited by 18 publications

References 21 publications

The second and third Sonine coefficients of a freely cooling granular gas revisited

The second and third Sonine coefficients of a freely cooling granular gas revisited

Modified Sonine approximation for the Navier–Stokes transport coefficients of a granular gas

Fluctuations of Power Injection in Randomly Driven Granular Gases

Contact Info

Product

Resources

About