Flowing Granular Materials and the Maxwell-Boltzmann Velocity Distribution

2000 ◽  
Author(s):  
Edward J. Boyle
Keyword(s):  
Distribution Function ◽  
Velocity Distribution ◽  
Hard Sphere ◽  
Body Force ◽  
Canonical Ensemble ◽  
Granular Flows ◽  
Velocity Distribution Function ◽  
Distribution Functions ◽  
Velocity Distribution Functions ◽  
Single Granule

Abstract The single-granule velocity distribution function is shown to be Maxwell-Boltzmann for hard-sphere granular flows at steady-state exhibiting no gradients and absent a body-force. This is accomplished by approximating the two-granule velocity distribution function as the product of two single-granule velocity distribution functions and a correlating function and by applying to a canonical ensemble a function analogous to Boltzmann’s H-function.

Statistical Inferences of Lift-Off Velocity Distribution Function Form for Saltating Particles

2010 ◽  
Vol 108-111 ◽  
pp. 783-788
Author(s):  
Jian Jun Wu ◽  
Li Hong He
Keyword(s):  
Distribution Function ◽  
Weibull Distribution ◽  
Velocity Distribution ◽  
Goodness Of Fit ◽  
Velocity Distribution Function ◽  
Forward Direction ◽  
Distribution Functions ◽  
Velocity Distribution Functions ◽  
Lift Off ◽  
Log Normal

The lift-off velocity distribution of saltating particles, which have been proposed to characterize the dislodgement state of saltating particles, is one of the key issues in the theoretical study of windblown sand transportation. But there were various statistical relations in the early researches. In this paper, the Kolmogorov-Smirnov test for goodness-of-fit is adopted to make an inference of the most probable form of lift-off velocity distribution functions for saltating particles on the basis of the experimental data. The statistical results show that the distribution function of vertical lift-off velocities conforms better to Weibull distribution function than to the normal, log-normal, gamma and exponential ones; while, the distribution function of the absolute values of horizontal lift-off velocities is best described by log-normal distribution in forward direction and Weibull distribution in backward direction, respectively. Finally, two more examples prove to support the above conclusions.

scholarly journals Velocity Distribution Functions and Transport Coefficients of Electron Swarms in Gases in the Presence of Crossed Electric and Magnetic Fields

1995 ◽  
Vol 48 (3) ◽  
pp. 557 ◽  
Author(s):  
KF Ness
Keyword(s):  
Distribution Function ◽  
Magnetic Fields ◽  
Velocity Distribution ◽  
Transport Coefficients ◽  
Velocity Distribution Function ◽  
Distribution Functions ◽  
Electron Motion ◽  
Electric And Magnetic Fields ◽  
Velocity Distribution Functions ◽  
Spatially Homogeneous

A multi-term solution of the Boltzmann equation is used to calculate the spatially homogeneous velocity distribution function of a dilute swarm of electrons moving through a background of denser neutral molecules in the presence of crossed electric and magnetic fields. As an example, electron motion in methane is considered.

Electromagnetic ion-cyclotron wave growth rates and their variation with velocity distribution function shape

1977 ◽  
Vol 17 (1) ◽  
pp. 123-131 ◽  
Author(s):  
Abraham Shrauner ◽  
W. C. Feldman
Keyword(s):  
Distribution Function ◽  
Velocity Distribution ◽  
Growth Rates ◽  
Velocity Distribution Function ◽  
Distribution Functions ◽  
Wave Growth ◽  
Ion Cyclotron ◽  
Velocity Distribution Functions ◽  
Velocity Moments ◽  
Function Shape

The sensitivity of electromagnetic ion-cyclotron wave growth rates to the details of the shape of proton velocity distribution functions is explored. For this purpose two different forms of bi-Lorentzian for the proton distribution functions were adopted. The growth rates for the two types of bi-Lorentzians and the biMaxwellians for the beam (hot) protons are compared. Although the growth rates for the three shapes depend on the velocity moments of the different velocity distributions in a similar way, their magnitudes were found to vary considerably.

scholarly journals The auroral O<sup>+</sup> non-Maxwellian velocity distribution function revisited

1997 ◽  
Vol 15 (2) ◽  
pp. 249-254 ◽  
Author(s):  
D. Hubert ◽  
F. Leblanc
Keyword(s):  
Magnetic Field ◽  
Distribution Function ◽  
Velocity Distribution ◽  
Polar Angle ◽  
Velocity Distribution Function ◽  
Distribution Functions ◽  
Field Direction ◽  
Magnetic Field Direction ◽  
Velocity Distribution Functions ◽  
The Magnetic Field

Abstract. New characteristics of O+ ion velocity distribution functions in a background of atomic oxygen neutrals subjected to intense external electromagnetic forces are presented. The one dimensional (1-D) distribution function along the magnetic field displays a core-halo shape which can be accurately fitted by a two Maxwellian model. The Maxwellian shape of the 1-D distribution function around a polar angle of 21 ± 1° from the magnetic field direction is confirmed, taking into account the accuracy of the Monte Carlo simulations. For the first time, the transition of the O+ 1-D distribution function from a core halo shape along the magnetic field direction to the well-known toroidal shape at large polar angles, through the Maxwellian shape at polar angle of 21 ± 1° is properly explained from a generic functional of the velocity moments at order 2 and 4.

scholarly journals On the Determination of Kappa Distribution Functions from Space Plasma Observations

Entropy ◽  
2020 ◽  
Vol 22 (2) ◽  
pp. 212 ◽  
Author(s):  
Georgios Nicolaou ◽  
George Livadiotis ◽  
Robert T. Wicks
Keyword(s):  
Distribution Function ◽  
Velocity Distribution ◽  
Space Plasma ◽  
Distribution Functions ◽  
Specific Method ◽  
Kappa Distribution ◽  
Kappa Index ◽  
Velocity Distribution Functions ◽  
Bulk Parameters ◽  
Plasma Bulk

The velocities of space plasma particles, often follow kappa distribution functions. The kappa index, which labels and governs these distributions, is an important parameter in understanding the plasma dynamics. Space science missions often carry plasma instruments on board which observe the plasma particles and construct their velocity distribution functions. A proper analysis of the velocity distribution functions derives the plasma bulk parameters, such as the plasma density, speed, temperature, and kappa index. Commonly, the plasma bulk density, velocity, and temperature are determined from the velocity moments of the observed distribution function. Interestingly, recent studies demonstrated the calculation of the kappa index from the speed (kinetic energy) moments of the distribution function. Such a novel calculation could be very useful in future analyses and applications. This study examines the accuracy of the specific method using synthetic plasma proton observations by a typical electrostatic analyzer. We analyze the modeled observations in order to derive the plasma bulk parameters, which we compare with the parameters we used to model the observations in the first place. Through this comparison, we quantify the systematic and statistical errors in the derived moments, and we discuss their possible sources.

The velocity distribution function within a shock wave

1967 ◽  
Vol 30 (3) ◽  
pp. 479-487 ◽  
Author(s):  
G. A. Bird
Keyword(s):  
Shock Wave ◽  
Distribution Function ◽  
Velocity Distribution ◽  
Numerical Experiments ◽  
Equilibrium Distribution ◽  
Velocity Distribution Function ◽  
Distribution Functions ◽  
Rigid Sphere ◽  
Lateral Velocity ◽  
Shock Profile

The structure of normal shock waves in a gas composed of rigid sphere molecules is investigated by numerical experiments with a simulated gas on a digital computer. The non-equilibrium between the temperatures based on the longitudinal and lateral velocity components is studied and the results compared with the theory of Yen (1966). Details of the velocity distribution function are presented for a shock of Mach number 10. The distribution functions for both the longitudinal and lateral velocity components are plotted for a number of locations in the shock profile and are compared with the equilibrium distribution.

A study of the mass ratio dependence of the mixed species collision integral for isotropic velocity distribution functions

1982 ◽  
Vol 27 (1) ◽  
pp. 135-148 ◽  
Author(s):  
A. J. M. Garrett
Keyword(s):  
Distribution Function ◽  
Velocity Distribution ◽  
Particle Distribution ◽  
Velocity Distribution Function ◽  
Collision Integral ◽  
Distribution Functions ◽  
Velocity Space ◽  
Particle Distribution Function ◽  
Collision Term ◽  
Test Species

This paper is concerned with the Boltzmann collision integral for the one-particle distribution function of a test species of particle undergoing elastic collisions with particles of a second species which is in thermal equilibrium. This expression is studied as a function of the ratio of the masses of the test and host particles for the case when the test particle distribution function is isotropic in velocity space. The analysis can also be considered as referring to the zeroth-order spherical harmonic in velocity space of a general velocity distribution function. The resulting collision term, due originally to Davydov, is of Fokker–Planck form and effectively describes a diffusion in energy. The method of derivation employed here is more systematic than hitherto, and is used to calculate the first correction to the Davydov term. Differences between classical and quantum cross-sections are considered; the correction to the Davydov term is checked by means of a comparison with the exact solution of the associated eigenvalue problem for the special case of Maxwell interactions treated classically.

scholarly journals VIOLATION OF MOLECULAR CHAOS IN DISSIPATIVE GASES

2002 ◽  
Vol 13 (09) ◽  
pp. 1263-1272 ◽  
Author(s):  
THORSTEN PÖSCHEL ◽  
NIKOLAI V. BRILLIANTOV ◽  
THOMAS SCHWAGER
Keyword(s):  
Distribution Function ◽  
Velocity Distribution ◽  
Numerical Simulations ◽  
Decay Rate ◽  
Particle Velocity ◽  
Hard Sphere ◽  
Velocity Distribution Function ◽  
Inelastic Collisions ◽  
Temperature Decay ◽  
Particle Velocities

Numerical simulations of a dissipative hard sphere gas reveal a dependence of the cooling rate on correlation of the particle velocities due to inelastic collisions. We propose a coefficient which characterizes the velocity correlations in the two-particle velocity distribution function and express the temperature decay rate in terms of this coefficient. The analytical results are compared with numerics.

High frequency oscillations in a bounded thermally produced plasma

1978 ◽  
Vol 360 (1703) ◽  
pp. 541-555 ◽  
Keyword(s):  
Velocity Distribution ◽  
High Frequency ◽  
Velocity Distribution Function ◽  
Distribution Functions ◽  
Electron Velocity ◽  
Electron Velocity Distribution ◽  
Electron Velocity Distribution Function ◽  
High Frequency Mode ◽  
High Frequency Oscillations ◽  
Velocity Distribution Functions

Measurements are reported of a high frequency mode of oscillation of a current carrying, bounded, thermally produced plasma. The oscillation occurs at approximately a third of the electron plasma frequency. A linear theory is presented, which predicts an instability at this frequency. The electron and ion velocity distribution functions used in this theory are determined by the self-consistent d.c. potential distribution. An interesting feature is that the electron velocity distribution function is a discontinuous Maxwellian. The calculated frequency of the instability and its variation as a function of the plasma density agree very well with the experimental observations.

scholarly journals A Quadrature-Based Kinetic Model for Dilute Non-Isothermal Granular Flows

2011 ◽  
Vol 10 (1) ◽  
pp. 216-252 ◽  
Author(s):  
Alberto Passalacqua ◽  
Janine E. Galvin ◽  
Prakash Vedula ◽  
Christine M. Hrenya ◽  
Rodney O. Fox
Keyword(s):  
Distribution Function ◽  
Velocity Distribution ◽  
Hard Sphere ◽  
Gaussian Quadrature ◽  
Finite Size ◽  
Velocity Distribution Function ◽  
Md Simulations ◽  
Collision Integral ◽  
Collision Kernel ◽  
Quadrature Formulas

AbstractA moment method with closures based on Gaussian quadrature formulas is proposed to solve the Boltzmann kinetic equation with a hard-sphere collision kernel for mono-dispersed particles. Different orders of accuracy in terms of the moments of the velocity distribution function are considered, accounting for moments up to seventh order. Quadrature-based closures for four different models for inelastic collision-the Bhatnagar-Gross-Krook, ES-BGK, the Maxwell model for hard-sphere collisions, and the full Boltzmann hard-sphere collision integral-are derived and compared. The approach is validated studying a dilute non-isothermal granular flow of inelastic particles between two stationary Maxwellian walls. Results obtained from the kinetic models are compared with the predictions of molecular dynamics (MD) simulations of a nearly equivalent system with finite-size particles. The influence of the number of quadrature nodes used to approximate the velocity distribution function on the accuracy of the predictions is assessed. Results for constitutive quantities such as the stress tensor and the heat flux are provided, and show the capability of the quadrature-based approach to predict them in agreement with the MD simulations under dilute conditions.

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