Velocity distribution of fluidized granular gases in the presence of gravity

Granular dynamics and gravity

The role of gravity on the dynamics of granular particles is examined via their velocity distributions. Acceleration due to gravity, particle number and the coefficient of restitution have all been varied. Various quantitative relationships governing the relationship between parameters involved in the vertical and horizontal velocity component distributions are derived and applied to the data, and for the horizontal velocity components' distribution in particular remarkably good agreement is obtained.

Velocity statistics of dynamic spinners in out-of-equilibrium magnetic suspensions

We report on the velocity statistics of an out-of-equilibrium magnetic suspension in a spinner phase confined at a liquid interface. The suspension is energized by a uniaxial alternating magnetic field applied parallel to the interface. In a certain range of the magnetic field parameters the system spontaneously undergoes a transition into a dynamic spinner phase (ensemble of hydrodynamically coupled magnetic micro-rotors) comprised of two subsystems: self-assembled spinning chains and a gas of rotating single particles. Both subsystems coexist in a dynamic equilibrium via continuous exchange of the particles. Spinners excite surface flows that significantly increase particle velocity correlations in the system. For both subsystems the velocity distributions are strongly non-Maxwellian with nearly exponential high-energy tails, P(v) ∼ exp(-|v/v0|). The kurtosis, the measure of the deviation from the Gaussian statistics, is influenced by the frequency of the external magnetic field. We show that in the single-particle gas the dissipation is mostly collisional, whereas the viscous damping dominates over collisional dissipation for the self-assembled spinners. The dissipation increases with the frequency of the applied magnetic field. Our results provide insights into non-trivial dissipation mechanisms determining self-assembly processes in out-of-equilibrium magnetic suspensions.

Self-organized shocks in the sedimentation of a granular gas

A granular gas in gravity heated from below develops a certain stationary density profile. When the heating is switched off, the granular gas collapses. We investigate the process of sedimentation using computational hydrodynamics, based on the Jenkins-Richman theory, and find that the process is significantly more complex than generally acknowledged. In particular, during its evolution, the system passes several stages which reveal distinct spatial regions of inertial (supersonic) and diffusive (subsonic) dynamics. During the supersonic stages, characterized by Mach>1, the system develops supersonic shocks which are followed by a steep front of the hydrodynamic fields of temperature and density, traveling upward.

Phase transitions in vibrated granular systems in microgravity

We numerically investigated various dynamical behaviors of a vibrated granular gas in microgravity. Using the parameters of an earlier Mini-Texus 5 experiment, three-dimensional simulations, based on molecular dynamics, efficiently reproduce experimental results. Using Kolmogorov-Smirnov tests, four dynamical regimes have been distinguished: gaseous state, partial clustering, complete clustering, and bouncing aggregates. Different grain radii and densities have been considered in order to describe a complete (r,η)-phase diagram. The latter exhibits rich features such as phase transitions and triple points. Our work emphasizes the complexity of diluted granular systems and opens fundamental perspectives.

Particle kinematics in a dilute, three-dimensional, vibration-fluidized granular medium

We report an experimental study of particle kinematics in a three-dimensional system of inelastic spheres fluidized by intense vibration. The motion of particles in the interior of the medium is tracked by high-speed video imaging, yielding a spatially resolved measurement of the velocity distribution. The distribution is wider than a Gaussian and broadens continuously with increasing volume fraction. The deviations from a Gaussian distribution for this boundary-driven system are different in sign and larger in magnitude than predictions for homogeneously driven systems. We also find correlations between velocity components which grow with increasing volume fraction.

Boltzmann equation for dissipative gases in homogeneous states with nonlinear friction

Combining analytical and numerical methods, we study within the framework of the homogeneous nonlinear Boltzmann equation a broad class of models relevant for the dynamics of dissipative fluids, including granular gases. We use the method presented in a previous paper [J. Stat. Phys. 124, 549 (2006)] and extend our results to a different heating mechanism: namely, a deterministic nonlinear friction force. We derive analytically the high-energy tail of the velocity distribution and compare the theoretical predictions with high-precision numerical simulations. Stretched exponential forms are obtained when the nonequilibrium steady state is stable. We derive subleading corrections and emphasize their relevance. In marginal stability cases, power-law behaviors arise, with exponents obtained as the roots of transcendental equations. We also consider some simple Bhatnagar-Gross-Krook models, driven by similar heating devices, to test the robustness of our predictions.

Velocity statistics in dissipative, dense granular media

We use a two-dimensional random-force model to investigate the velocity distributions in driven granular media. In general, the shape of the distribution is found to depend on the degree of dissipation and the packing fraction but, in highly dissipative systems, the velocity distributions have tails close to exponential. We show that these arise from the dynamics of single particles traveling in dilute regions and influenced predominantly by the random force. A self-consistent kinetic theory is developed to describe this behavior.

Velocity distributions of granular gases with drag and with long-range interactions

We study velocity statistics of electrostatically driven granular gases. For two different experiments, (i) nonmagnetic particles in a viscous fluid and (ii) magnetic particles in air, the velocity distribution is non-Maxwellian, and its high-energy tail is exponential, P(upsilon) approximately exp(-/upsilon/). This behavior is consistent with the kinetic theory of driven dissipative particles. For particles immersed in a fluid, viscous damping is responsible for the exponential tail, while for magnetic particles, long-range interactions cause the exponential tail. We conclude that velocity statistics of dissipative gases are sensitive to the fluid environment and to the form of the particle interaction.

Competitive clustering in a bidisperse granular gas: experiment, molecular dynamics, and flux model

A compartmentalized bidisperse granular gas clusters competitively [Phys. Rev. Lett. 89, 214301 (2002)]]: By tuning the shaking strength, the clustering can be directed either towards the compartment initially containing mainly small particles or to the compartment containing mainly large particles. Here, the conditions under which this competitive clustering occurs are studied experimentally, numerically (by means of molecular dynamics simulations), and analytically. A minimal model is derived that quantitatively accounts for the observed phenomena.

Local equation of state and velocity distributions of a driven granular gas

We present event-driven simulations of a granular gas of inelastic hard disks with incomplete normal restitution in two dimensions between vibrating walls (without gravity). We measure hydrodynamic quantities such as the stress tensor, density and temperature profiles, as well as velocity distributions. Relating the local pressure to the local temperature and local density, we construct a local constitutive equation. For strong inelasticities the local constitutive relation depends on global system parameters, like the volume fraction and the aspect ratio. For moderate inelasticities the constitutive relation is approximately independent of the system parameters and can hence be regarded as a local equation of state, even though the system is highly inhomogeneous with heterogeneous temperature and density profiles arising as a consequence of energy injection. With respect to local velocity distributions we find that they do not scale with the square root of the local granular temperature. Moreover the high-velocity tails are different for the distribution of the x and the y components of the velocity, and even depend on the position in the sample, the global volume fraction, and the coefficient of restitution.

Crucial role of sidewalls in velocity distributions in quasi-two-dimensional granular gases

Our experiments and three-dimensional molecular dynamics simulations of particles confined to a vertical monolayer by closely spaced frictional walls (sidewalls) yield velocity distributions with non-Gaussian tails and a peak near zero velocity. Simulations with frictionless sidewalls are not peaked. Thus interactions between particles and their containers are an important determinant of the shape of the distribution and should be considered when evaluating experiments on a constrained monolayer of particles.

Gaussian kinetic model for granular gases

A kinetic model for the Boltzmann equation is proposed and explored as a practical means to investigate the properties of a dilute granular gas. It is shown that all spatially homogeneous initial distributions approach a universal "homogeneous cooling solution" after a few collisions. The homogeneous cooling solution (HCS) is studied in some detail and the exact solution is compared with known results for the hard sphere Boltzmann equation. It is shown that all qualitative features of the HCS, including the nature of overpopulation at large velocities, are reproduced by the kinetic model. It is also shown that all the transport coefficients are in excellent agreement with those from the Boltzmann equation. Also, the model is specialized to one having a velocity independent collision frequency and the resulting HCS and transport coefficients are compared to known results for the Maxwell model. The potential of the model for the study of more complex spatially inhomogeneous states is discussed.

NMR experiments on a three-dimensional vibrofluidized granular medium

A three-dimensional granular system fluidized by vertical container vibrations was studied using pulsed field gradient NMR coupled with one-dimensional magnetic resonance imaging. The system consisted of mustard seeds vibrated vertically at 50 Hz, and the number of layers N(l)<or=4 was sufficiently low to achieve a nearly time-independent granular fluid. Using NMR, the vertical profiles of density and granular temperature were directly measured, along with the distributions of vertical and horizontal grain velocities. The velocity distributions showed modest deviations from Maxwell-Boltzmann statistics, except for the vertical velocity distribution near the sample bottom, which was highly skewed and non-Gaussian. Data taken for three values of N(l) and two dimensionless accelerations Gamma=15,18 were fitted to a hydrodynamic theory, which successfully models the density and temperature profiles away from the vibrating container bottom. A temperature inversion near the free upper surface is observed, in agreement with predictions based on the hydrodynamic parameter micro which is nonzero only in inelastic systems.

Steady-state velocity distributions of an oscillated granular gas

We use a three-dimensional molecular dynamics simulation to study the single particle distribution function of a dilute granular gas driven by a vertically oscillating plate at high accelerations (15g-90g). We find that the density and the temperature fields are essentially time-invariant above a height of about 40 particle diameters, where typically 20% of the grains are contained. These grains form the nonequilibrium steady-state granular gas with a Knudsen number unity or greater. In the steady-state region, the probability distribution function of the horizontal velocity c(x) (scaled by the local horizontal temperature) is found to be nearly independent of height, even though the hydrodynamic fields vary with height. We find that the high energy tails of the distribution function are described by a stretched exponential approximately exp(-Bcalphax), where alpha depends on the restitution coefficient e and falls in the range 1.2<alpha<1.6. However, alpha does not vary significantly for a wide range of friction coefficient values. We find that the distribution function of a frictionless inelastic hard sphere model can be made similar to that of a frictional model by adjusting e. However, there is no single value of e that mimics the frictional model over a range of heights.

Fluid-like behavior of a one-dimensional granular gas

We study the properties of a one-dimensional (1D) granular gas consisting of N hard rods on a line of length L (with periodic boundary conditions). The particles collide inelastically and are fluidized by a heat bath at temperature Tb and viscosity gamma. The analysis is supported by molecular dynamics simulations. The average properties of the system are first discussed, focusing on the relations between granular temperature Tg=mv2, kinetic pressure, and density rho=N/L. Thereafter, we consider the fluctuations around the average behavior obtaining a slightly non-Gaussian behavior of the velocity distributions and a spatially correlated velocity field; the density field displays clustering: this is reflected in the structure factor which has a peak in the k approximately 0 region suggesting an analogy between inelastic hard core interactions and an effective attractive potential. Finally, we study the transport properties, showing the typical subdiffusive behavior of 1D stochastically driven systems, i.e., </x(t)-x(0)/2> approximately Dt(1/2), where D for the inelastic fluid is larger than the elastic case. This is directly related to the peak of the structure factor at small wave vectors.

Collision statistics of driven granular materials

We present an experimental investigation of the statistical properties of spherical granular particles on an inclined plane that are excited by an oscillating side wall. The data is obtained by high-speed imaging and particle tracking techniques. We identify all particles in the system and link their positions to form trajectories over long times. Thus, we identify particle collisions to measure the effective coefficient of restitution and find a broad distribution of values for the same impact angles. We find that the energy inelasticity can take on values greater than one, which implies that the rotational degrees of freedom play an important role in energy transfer. We also measure the distance and the time between collision events in order to directly determine the distribution of path lengths and the free times. These distributions are shown to deviate from expected theoretical forms for elastic spheres, demonstrating the inherent clustering in this system. We describe the data with a two-parameter fitting function and use it to calculate the mean free path and collision time. We find that the ratio of these values is consistent with the average velocity. The velocity distributions are observed to be strongly non-Gaussian and do not demonstrate any apparent universal behavior. We report the scaling of the second moment, which corresponds to the granular temperature, and higher order moments as a function of distance from the driving wall. Additionally, we measure long-time correlation functions in both space and in the velocities to probe diffusion in a dissipative gas.