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

We consider a dilute granular gas of hard spheres colliding inelastically with coefficients of normal and tangential restitution ␣ and ␤, respectively. The basic quantities characterizing the distribution function f͑v , ͒ of linear ͑v͒ and angular ͑͒ velocities are the second-degree moments defining the translational ͑T tr ͒ and rotational ͑T rot ͒ temperatures. The deviation of f from the Maxwellian distribution parameterized by T tr and T rot can be measured by the cumulants associated with the fourth-degree velocity moments. The main objective of this paper is the evaluation of the collisional rates of change of these second-and fourth-degree moments by means of a Sonine approximation. The results are subsequently applied to the computation of the temperature ratio T rot / T tr and the cumulants of two paradigmatic states: the homogeneous cooling state and the homogeneous steady state driven by a white-noise stochastic thermostat. It is found in both cases that the Maxwellian approximation for the temperature ratio does not deviate much from the Sonine prediction. On the other hand, non-Maxwellian properties measured by the cumulants cannot be ignored, especially in the homogeneous cooling state for medium and small roughness. In that state, moreover, the cumulant directly related to the translational velocity differs in the quasi-smooth limit ␤ → −1 from that of pure smooth spheres ͑␤ =−1͒. This singular behavior is directly related to the unsteady character of the homogeneous cooling state and thus it is absent in the stochastic thermostat case.

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“…The method will be similar to that already worked out in the case of smooth spheres. [41][42][43] FIG. 1.…”

Section: ͑114͒mentioning

confidence: 96%

Sonine approximation for collisional moments of granular gases of inelastic rough spheres

2011

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“…However, if inelasticity is not too large, the nonlinear contributions of a 2 and the complete contributions of higher order cumulants can be neglected. This is the so-called first Sonine approximation [56,60], which yields μ n ≃ μ ð0Þ n þ μ ð1Þ n a 2 , with μ (1), they become a closed set, but the T-a 2 coupling still remains. Taking into account this coupling, and since μ 2 is an increasing function of a 2 , it turns out that the relaxation of the granular temperature T from an initially "cooler" (smaller T) sample could possibly be overtaken by that of an initially "hotter" one, if the initial excess kurtosis of the latter is larger enough.…”

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confidence: 99%

When the Hotter Cools More Quickly: Mpemba Effect in Granular Fluids

et al. 2017

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Under certain conditions, two samples of fluid at different initial temperatures present a counterintuitive behavior known as the Mpemba effect: it is the hotter system that cools sooner. Here, we show that the Mpemba effect is present in granular fluids, both in uniformly heated and in freely cooling systems. In both cases, the system remains homogeneous, and no phase transition is present. Analytical quantitative predictions are given for how differently the system must be initially prepared to observe the Mpemba effect, the theoretical predictions being confirmed by both molecular dynamics and Monte Carlo simulations. Possible implications of our analysis for other systems are also discussed. DOI: 10.1103/PhysRevLett.119.148001 Let us consider two identical beakers of water, initially at two different temperatures, put in contact with a thermal reservoir at subzero (on the Celsius scale) temperature. While one may intuitively expect that the initially cooler sample would freeze first, it has been observed that this is not always the case [1]. This paradoxical behavior named the Mpemba effect (ME) has been known since antiquity and discussed by philosophers like Aristotle, Roger Bacon, Francis Bacon, and Descartes [2,3]. Nevertheless, physicists only started to analyze it in the second part of the past century, mainly in popular science or education journals .There is no consensus on the underlying physical mechanisms that bring about the ME. Specifically, water evaporation [4,5,9,24], differences in the gas composition of water [11,17,25], natural convection [6,23,26], or the influence of supercooling, either alone [14,27] or combined with other causes [28][29][30][31], have been claimed to have an impact on the ME. Conversely, the own existence of the ME in water has been recently put in question [32]. Notwithstanding, Mpemba-like effects have also been observed in different physical systems, such as carbon nanotube resonators [33] or clathrate hydrates [34].The ME requires the evolution equation for the temperature to involve other variables, which may facilitate or hinder the temperature relaxation rate. The initial values of those additional variables depend on the way the system has been prepared, i.e., "aged," before starting the relaxation process. Typically, aging and memory effects are associated with slowly evolving systems with a complex energy landscape, such as glassy [35][36][37][38][39][40][41][42][43] or dense granular systems [44][45][46]. However, these effects have also been observed in simpler systems, like granular gases [47][48][49][50] or, very recently, crumpled thin sheets and elastic foams [51].In a general physical system, the study of the ME implies finding those additional variables that control the temperature relaxation and determining how different they have to be initially in order to facilitate its emergence. In order to quantify the effect with the tools of nonequilibrium statistical mechanics, a precise definition thereof is mandatory. An option is to look at the relaxat...

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“…(6.9) is independent of the parameters β and γ. In fact, when ξ = 0, one recovers the kinetic equation defining the homogeneous cooling state (HCS), whose solution has been previously worked out by several authors Montanero & Santos 2000;Pöschel & Brilliantov 2006;Santos & Montanero 2009). In terms of the (scaled) distribution Ψ, Eq.…”

Section: Local Homogeneous State Zeroth-order Solutionmentioning

confidence: 99%

Development, Verification, and Validation of Multiphase Models for Polydisperse Flows

Hrenya¹,

Cocco²,

Fox³

et al. 2011

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This report describes in detail the technical findings of the DOE Award entitled "Development, Verification, and Validation of Multiphase Models for Polydisperse Flows." The focus was on high-velocity, gas-solid flows with a range of particle sizes. A complete mathematical model was developed based on first principles and incorporated into MFIX. The solid-phase description took two forms: the Kinetic Theory of Granular Flows (KTGF) and Discrete Quadrature Method of Moments (DQMOM). The gas-solid drag law for polydisperse flows was developed over a range of flow conditions using Discrete Numerical Simulations (DNS). These models were verified via examination of a range of limiting cases and comparison with Discrete Element Method (DEM) data. Validation took the form of comparison with both DEM and experimental data. Experiments were conducted in three separate circulating fluidized beds (CFB's), with emphasis on the riser section. Measurements included bulk quantities like pressure drop and elutriation, as well as axial and radial measurements of bubble characteristics, cluster characteristics, solids flux, and differential pressure drops (axial only). Monodisperse systems were compared to their binary and continuous particle size distribution (PSD) counterparts. The continuous distributions examined included Gaussian, lognormal, and NETLprovided data for a coal gasifier.FINAL TECHNICAL 4 DE-FC26-07NT43098 5 EXECUTIVE SUMMARYThis report is the final technical report for the award DE-FC26-07NT43098 entitled "Development, Verification, and Validation of Multiphase Models for Polydisperse Flows." Below is a summary of work completed under the grant for each of the major goals. Goal I: Continuum Theory for Solid PhaseTwo separate and complementary continuum theories were derived to model polydisperse solids: the kinetic theory of granular flow (KTGF) and the discrete quadrature method of moments (DQMOM). Both theories are based on first principles with no adjustable parameters. The polydisperse KTGF and DQMOM were then encoded in MFIX, and underwent a wide range of verification testing to ensure, to the best possible extent, that no coding errors were present. These verification tests included both granular and gas-solid flows, including cases where an analytical solution and/or DEM (discrete element method) data and/or simple test cases. For the case of KTGF, another set of constitutive equations were derived which rigorously incorporated the gas phase for a simplified case of monodisperse systems at low Reynolds numbers. This derivation used the acceleration model developed in Task 2 from direct numerical simulations (DNS) in the starting kinetic equation, and indicated the effect of the fluid phase on the resulting constitutive relations. Goal II: Improved Gas-Particle Drag Laws -effect of particle size distributionIn this portion of the effort, two types of direct numerical simulations (DNS) were used to extract the drag force experienced by particles in a polydisperse suspension. These two methods, na...

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

Cited by 42 publications

References 28 publications

Sonine approximation for collisional moments of granular gases of inelastic rough spheres

Sonine approximation for collisional moments of granular gases of inelastic rough spheres

When the Hotter Cools More Quickly: Mpemba Effect in Granular Fluids

Development, Verification, and Validation of Multiphase Models for Polydisperse Flows

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