Granular Gas Research Articles

Jumping Platonic Solids on a Vibrating Plate

AbstractThe mechanical excitation of solid bodies by a vibrating plate is not only an interesting fundamental problem, but also has relevance in many mechanical systems, for example, in the preparation of granular gases in microgravity. Herein, the energy input by an oscillating plate is numerically investigated as a function of the excitation parameters for selected Platonic solids and their behavior compared to that of a jumping sphere. The most important additional features, not relevant for spheres, are the excitation of body rotations and a permanent energy exchange between the rotational and translational degrees of freedom during the collisions with the plate. The distribution of kinetic energies is analyzed by a numerical simulation of the dynamics, using structures which emulate the mechanical behavior of regular polyhedra.

Do consumers pay the corporate tax?

AbstractUsing granular gas price data and rich variation in corporate tax rates, we find that corporate taxes increase consumer prices. About 64% of the corporate tax is borne by consumers. The effect is stronger when firms have limited access to tax planning opportunities, face stricter tax enforcement, or when consumer demand is less elastic. Taxes also reduce the number of firms and their scale, consistent with a tax‐induced increase in marginal cost. Our results suggest that tax policies that increase effective corporate tax rates may have unintended consequences for consumers through higher prices.

Моделирование обрушения горной породы

Introduction. At present, the study of the movement of falling rock (collapses of rocks), the causes of their occurrence, as well as their influence on other slope processes is an important field of research. These studies are based on both theoretical knowledge and experimental observations. Separately, it is necessary to highlight the methods of mathematical and computer modeling, which are widely used in cases where experimental data are not enough. The study of the influence of the slope angle and the slope height (on which the falling rock mass was located at the initial moment of time) on the affected area was made using the methods of mathematical and computer modeling. Materials and methods. Two-fluid model based on the continuum approach and the kinetic theory of granular gas was used for a theoretical study of the movement of falling rock along the slope. The model takes into account the fluidization of the falling mass flow, and its implementation does not require the use of powerful computing resources. The modeling results obtained using the two-fluid model were compared with the results of an experimental study on the collapse of dolomite particles (average diameter 5 mm) carried out in the laboratory. The values of the slope height varied from 35 cm to 86 cm, the values of the slope angle varied from 35 degrees to 53 degrees in accordance with the experimental data. The results and discussion. The graphs of the distance of the rock run after its collapse in the dependence of the slope height and the slope angle were plotted. It is found that the calculation results are in good agreement with the experimental data (the relative error does not exceed 11%) for relatively large values of the slope angle. The results of calculations of the distance of the rock run are somewhat overestimated relative to the experimental data for small values of the slope angle. It is found that the affected area of dolomite particles increases with increasing slope angle and slope height. It should be noted that the affected area depends more on the values of the slope angle than on the height of the slope. If the slope is steeper (the value of the slope angle is greater), than the affected area is greater. Conclusion. The calculation results obtained using the two-fluid model describe the experimental data of the rock collapse satisfactorily. The results of calculations of the distance of the rock run are in good agreement with the experimental data (the relative error does not exceed 11%) for relatively large values of the slope angle. It is found that the distance of the rock run after its collapse depends more on the values of the slope angle than on the height of the slope. Resume. The research results can be useful in estimating the distance of the rock run after its collapse, as well as in developing a methodology for assessing affected areas during rock falls. The direction of future research is to improve the two-fluid model based on the continuum approach and the kinetic theory of granular gas for describing of real rock falls.

Diffusion of intruders in granular suspensions: Enskog theory and random walk interpretation.

The Enskog kinetic theory is applied to compute the mean square displacement of impurities or intruders (modeled as smooth inelastic hard spheres) immersed in a granular gas of smooth inelastic hard spheres (grains). Both species (intruders and grains) are surrounded by an interstitial molecular gas (background) that plays the role of a thermal bath. The influence of the latter on the motion of intruders and grains is modeled via a standard viscous drag force supplemented by a stochastic Langevin-like force proportional to the background temperature. We solve the corresponding Enskog-Lorentz kinetic equationby means of the Chapman-Enskog expansion truncated to first order in the gradient of the intruder number density. The integral equationfor the diffusion coefficient is solved by considering the first two Sonine approximations. To test these results, we also compute the diffusion coefficient from the numerical solution of the inelastic Enskog equationby means of the direct simulation Monte Carlo method. We find that the first Sonine approximation generally agrees well with the simulation results, although significant discrepancies arise when the intruders become lighter than the grains. Such discrepancies are largely mitigated by the use of the second Sonine approximation, in excellent agreement with computer simulations even for moderately strong inelasticities and/or dissimilar mass and diameter ratios. We invoke a random walk picture of the intruders' motion to shed light on the physics underlying the intricate dependence of the diffusion coefficient on the main system parameters. This approach, recently employed to study the case of an intruder immersed in a granular gas, also proves useful in the present case of a granular suspension. Finally, we discuss the applicability of our model to real systems in the self-diffusion case. We conclude that collisional effects may strongly impact the diffusion coefficient of the grains.

Revealing the Nonequilibrium Nature of a Granular Intruder: The Crucial Role of Non-Gaussian Behavior.

The characterization of the distance from equilibrium is a debated problem in particular in the treatment of experimental signals. If the signal is a one-dimensional time series, such a goal becomes challenging. A paradigmatic example is the angular diffusion of a rotator immersed in a vibro-fluidized granular gas. Here, we experimentally observe that the rotator's angular velocity exhibits significant differences with respect to an equilibrium process. Exploiting the presence of two relevant timescales and non-Gaussian velocity increments, we quantify the breakdown of time-reversal asymmetry, which would vanish in the case of a 1D Gaussian process. We deduce a new model for the massive probe, with two linearly coupled variables, incorporating both Gaussian and Poissonian noise, the latter motivated by the rarefied collisions with the granular bath particles. Our model reproduces the experiment in a range of densities, from dilute to moderately dense, with a meaningful dependence of the parameters on the density. We believe the framework proposed here opens the way to a more consistent and meaningful treatment of out-of-equilibrium and dissipative systems.

Mpemba effect in driven granular gases: Role of distance measures.

The Mpemba effect refers to the counterintuitive effect where a system which is initially further from the final steady state equilibrates faster than an identical system that is initially closer. The closeness to the final state is defined in terms of a distance measure. For driven granular systems, the Mpemba effect has been illustrated in terms of an ad hoc measure of mean kinetic energy as the distance function. In this paper, by studying four different distance measures based on the mean kinetic energies as well as velocity distribution, we show that the Mpemba effect depends on the definition of the measures.

Tracer diffusion coefficients in a moderately dense granular suspension: Stability analysis and thermal diffusion segregation

The diffusion transport coefficients of a binary granular suspension where one of the components is present in tracer concentration are determined from the (inelastic) Enskog kinetic equation. The effect of the interstitial gas on the solid particles is accounted for in the kinetic equation through two different terms: (i) a viscous drag force proportional to the particle velocity and (ii) stochastic Langevin-like term defined in terms of the background temperature. The transport coefficients are obtained as the solutions of a set of coupled linear integral equations recently derived for binary granular suspensions with arbitrary concentration [Gómez González et al., “Enskog kinetic theory for multicomponent granular suspensions,” Phys. Rev. E 101, 012904 (2020)]. To achieve analytical expressions for the diffusion coefficients, which can be sufficiently accurate for highly inelastic collisions and/or disparate values of the mass and diameter rations, the above integral equations are approximately solved by considering the so-called second Sonine approximation (two terms in the Sonine polynomial expansion of the distribution function). The theoretical results for the tracer diffusion coefficient D0 (coefficient connecting the mass flux with the gradient of density of tracer particles) are compared with those obtained by numerically solving the Enskog equation by means of the direct simulation Monte Carlo method. Although the first-Sonine approximation to D0 yields, in general, a good agreement with simulation results, we show that the second-Sonine approximation leads to an improvement over the first-Sonine correction, especially when the tracer particles are much lighter than the granular gas. The expressions derived here for the diffusion coefficients are also used for two different applications. First, the stability of the homogeneous steady state is discussed. Second, segregation induced by a thermal gradient is studied. As expected, the results show that the corresponding phase diagrams for segregation clearly differ from those found in previous works when the effect of gas phase on grains is neglected.

Translational and rotational non-Gaussianities in homogeneous freely evolving granular gases.

The importance of roughness in the modeling of granular gases has been increasingly considered in recent years. In this paper, a freely evolving homogeneous granular gas of inelastic and rough hard disks or spheres is studied under the assumptions of the Boltzmann kinetic equation. The homogeneous cooling state is studied from a theoretical point of view using a Sonine approximation, in contrast to a previous Maxwellian approach. A general theoretical description is done in terms of d_{t} translational and d_{r} rotational degrees of freedom, which accounts for the cases of spheres (d_{t}=d_{r}=3) and disks (d_{t}=2,d_{r}=1) within a unified framework. The non-Gaussianities of the velocity distribution function of this state are determined by means of the first nontrivial cumulants and by the derivation of non-Maxwellian high-velocity tails. The results are validated by computer simulations using direct simulation Monte Carlo and event-driven molecular dynamics algorithms.

Soft coarse-grained particle model for particle-fluid systems

By modeling a group of neighboring real particles as a single coarse-grained particle (CGP), discrete particle method (DPM) is now capable of simulating industrial-scale particle-fluid systems. However, a systematic approach to determine the CGP properties and develop their interaction models is still lacking, which casts uncertainty on the predictivity of the method. In this study, collisions between predefined particle groups are analyzed to construct kernel functions for modeling the CGPs and then the model parameters are determined by equating the statistical properties of the CGPs and the real particles in the physical process studied. This approach is implemented for homogeneous cooling of granular gas, then demonstrated effective in simulating experimental fluidized beds.

Lower Oxygen Tension and Intracranial Hemorrhage in Veno-venous Extracorporeal Membrane Oxygenation.

We examined the relationship between 24-h pre- and post-cannulation arterial oxygen tension (PaO2) and arterial carbon dioxide tension (PaCO2) and subsequent acute brain injury (ABI) in patients receiving veno-venous extracorporeal membrane oxygenation (VV-ECMO) with granular arterial blood gas (ABG) data and institutional standardized neuromonitoring. Eighty-nine patients underwent VV-ECMO (median age = 50, 63% male). Twenty (22%) patients experienced ABI; intracranial hemorrhage (ICH) was the most common diagnosis (n = 14, 16%). Lower post-cannulation PaO2 levels were significantly associated with ICH (66 vs. 81mmHg, p = 0.007) and a post-cannulation PaO2 level < 70mmHg was more frequent in these patients (71% vs. 33%, p = 0.007). PaCO2 parameters were not associated with ABI. By multivariable logistic regression, hypoxemia post-cannulation increased the odds of ICH (OR = 5.06, 95% CI:1.41-18.17; p = 0.01). In summary, lower oxygen tension in the 24-h post-cannulation was associated with ICH development. The precise roles of peri-cannulation ABG changes deserve further investigation, as they may influence the management of VV-ECMO patients.

Statistics of a two-dimensional immersed granular gas magnetically forced in volume.

We present an experimental study of the dynamics of a set of magnets within a fluid in which a remote torque applied by a vertical oscillating magnetic field transfers angular momentum to individual magnets. This system differs from previous experimental studies of granular gas where the energy is injected by vibrating the boundaries. Here, we do not observe any cluster formation, orientational correlation and equipartition of the energy. The magnets' linear velocity distributions are stretched exponentials, similar to three-dimensional boundary-forced dry granular gas systems, but the exponent does not depend on the number of magnets. The value of the exponent of the stretched exponential distributions is close to the value of 3/2 previously derived theoretically. Our results also show that the conversion rate of angular momentum into linear momentum during the collisions controls the dynamics of this homogenously forced granular gas. We report the differences among this homogeneously forced granular gas, ideal gas, and nonequilibrium boundary-forced dissipative granular gas.

Universal cover-time distributions of random motion in bounded granular gases.

The exhaustive random exploration of a complex domain is a fundamental issue in many natural, social, and engineering systems. The key characterizing quantity is the cover time, which is the time to visit every site in the system. One prototypical experimental platform is the confined granular gas, where the random motion of granular particles mimics the wandering of random walkers in a confined region. Here, we investigate the cover-time distribution of the random motion of tracer particles in granular gases confined in four containers to account for different boundary and angle effects and examine whether the cover time of the heterogeneous random motion of the granular gases can be rescaled into the universal Gumbel distribution according to a recent theory [Dong et al., arXiv:2210.05122 (2022)]. It is found that for long cover times, the experimental results are in full accord, while for short cover times, the agreement is reasonable, with noticeable deviations that can be attributed to spatial correlations of the sites in the covering process. Our results, thus, call for further theoretical investigations in order to take into full account these nonideal issues.

High-Degree Collisional Moments of Inelastic Maxwell Mixtures—Application to the Homogeneous Cooling and Uniform Shear Flow States

The Boltzmann equation for d-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.

Rheology of dilute granular gas mixtures where the grains interact via a square shoulder and well potential

We develop the rheology of a dilute granular gas mixture. Motivated by the interaction of charged granular particles, we assume that the grains interact via a square shoulder and well potential. Employing a kinetic theory, we compute the temperature and the shear viscosity as a function of the shear rate. Numerical simulations confirm our results are above the critical shear rate. At a shear rate below a critical value, clustering of the particles occurs.

High-energy velocity tails in uniformly heated granular materials.

We experimentally investigate the velocity distributions of quasi-two-dimensional granular materials uniformly heated by an electromagnetic vibrator, where the translational velocity and the rotation of a single particle are Gaussian and independent. We observe the non-Gaussian distributions of particle velocity, with the density-independent high-energy tails characterized by an exponent of β=1.50±0.03 for volume fractions of 0.111≤ϕ≤0.832, covering a wide range of structures and dynamics. Surprisingly, our results are not only in excellent agreement with the prediction of the kinetic theories of granular gas but also hold for an extremely high-volume fraction of ϕ=0.832 where the granular material forms a crystalline solid and the kinetic theory of granular gas fails fantastically. Our experiment suggests that the density-independent high-energy velocity tails of β=1.50 are a characteristic of uniformly heated granular matter.

Computing the mean square displacement of intruders in freely cooling granular gases with kinetic theory and a random flight approach

Computing the mean square displacement of intruders in freely cooling granular gases with kinetic theory and a random flight approach

Kinetic Theory and Memory Effects of Homogeneous Inelastic Granular Gases under Nonlinear Drag.

We study a dilute granular gas immersed in a thermal bath made of smaller particles with masses not much smaller than the granular ones in this work. Granular particles are assumed to have inelastic and hard interactions, losing energy in collisions as accounted by a constant coefficient of normal restitution. The interaction with the thermal bath is modeled by a nonlinear drag force plus a white-noise stochastic force. The kinetic theory for this system is described by an Enskog–Fokker–Planck equation for the one-particle velocity distribution function. To get explicit results of the temperature aging and steady states, Maxwellian and first Sonine approximations are developed. The latter takes into account the coupling of the excess kurtosis with the temperature. Theoretical predictions are compared with direct simulation Monte Carlo and event-driven molecular dynamics simulations. While good results for the granular temperature are obtained from the Maxwellian approximation, a much better agreement, especially as inelasticity and drag nonlinearity increase, is found when using the first Sonine approximation. The latter approximation is, additionally, crucial to account for memory effects such as Mpemba and Kovacs-like ones.

Mpemba-like effect protocol for granular gases of inelastic and rough hard disks

We study the conditions under which a Mpemba-like effect emerges in granular gases of inelastic and rough hard disks driven by a class of thermostats characterized by the splitting of the noise intensity into translational and rotational counterparts. Thus, granular particles are affected by a stochastic force and a stochastic torque, which inject translational and rotational energy, respectively. We realize that a certain choice of a thermostat of this class can be characterized just by the total intensity and the fraction of noise transferred to the rotational degree of freedom with respect to the translational ones. Firstly, Mpemba effect is characterized by the appearance of a crossing between the temperature curves of the considered samples. Later, an overshoot of the temperature evolution with respect to the steady-state value is observed and the mechanism of Mpemba effect generation is changed. The choice of parameters allows us to design plausible protocols based on these thermostats for generating the initial states to observe the Mpemba-like effect in experiments. In order to obtain explicit results, we use a well-founded Maxwellian approximation for the evolution dynamics and the steady-state quantities. Finally, theoretical results are compared with direct simulation Monte Carlo and molecular dynamics results, and a very good agreement is found.

Human-Scale Brownian Ratchet: A Historical Thought Experiment.

We present an experimental realization at macroscopic scale of the storied Brownian ratchet, which is an illustration of the Maxwell's demon. In our mechanism, the rotation of a centimeter-scale 1D Brownian object in a granular gas is detected by an electromechanical converter (dynamo), generating a voltage proportional to its angular velocity. The current generated by this random rotation is rectified by an electronic device (demon), such that only positive current passes. Eventually, work can be produced. The advantage of such a macroscopic setup is to allow measurement of all the observables with time: useful power (work), heat taken from the bath, and finally the efficiency of the equivalent heat engine. The feedback allowing the conversion from heat into work expresses as a bias on the Brownian motion.

On the mean square displacement of intruders in freely cooling granular gases

We compute the mean square displacement (MSD) of intruders immersed in a freely cooling granular gas made up of smooth inelastic hard spheres. In general, intruders and particles of the granular gas are assumed to have different mechanical properties, implying that non-equipartition of energy must be accounted for in the computation of the diffusion coefficient D. In the hydrodynamic regime, the time decay of the granular temperature T of the cooling granular gas is known to be dictated by Haff’s law; the corresponding decay of the intruder’s collision frequency entails a time decrease of the diffusion coefficient D. Explicit knowledge of this time dependence allows us to determine the MSD by integrating the corresponding diffusion equation. As in previous studies of self-diffusion (intruders mechanically equivalent to gas particles) and the Brownian limit (intruder’s mass much larger than the grain’s mass), we find a logarithmic time dependence of the MSD as a consequence of Haff’s law. This dependence extends well beyond the two aforementioned cases, as it holds in all spatial dimensions for arbitrary values of the mechanical parameters of the system (masses and diameters of intruders and grains, as well as their coefficients of normal restitution). Our result for self-diffusion in a three-dimensional granular gas agrees qualitatively, but not quantitatively, with that recently obtained by Blumenfeld [arXiv: 2111.06260] in the framework of a random walk model. Beyond the logarithmic time growth, we find that the MSD depends on the mechanical system parameters in a highly complex way. We carry out a comprehensive analysis from which interesting features emerge, such a non-monotonic dependence of the MSD on the coefficients of normal restitution and on the intruder-grain mass ratio. To explain the observed behaviour, we analyze in detail the intruder’s random walk, consisting of ballistic displacements interrupted by anisotropic deflections caused by the collisions with the hard spheres. We also show that the MSD can be thought of as arising from an equivalent random walk with isotropic, uncorrelated steps. Finally, we derive some results for the MSD of an intruder inmersed in a driven granular gas and compare them with those obtained for the freely cooling case. In general, we find significant quantitative differences in the dependence of the scaled diffusion coefficient on the coefficient of normal restitution for the grain-grain collisions.Graphic abstract