Granular Gas Dynamics (Lecture Notes in Physics, 624)

✍ Scribed by Thorsten Pöschel (editor), Nikolai V. Brilliantov (editor)

• Publisher: Springer
• Year: 2003
• Tongue: English
• Leaves: 369
• Edition: 2003
• Category: Library

✦ Synopsis

While there is not yet any general theory for granular materials, significant progress has been achieved for dilute systems, also called granular gases. The contributions in this book address both the kinetic approach one using the Boltzmann equation for dissipative gases as well as the less established hydrodynamic description. The last part of the book is devoted to driven granular gases and their analogy with molecular fluids. Care has been taken so as to present the material in a pedagogical and self-contained way and this volume will thus be particularly useful to nonspecialists and newcomers to the field.

✦ Table of Contents

front-matter
Chapter 1
1 Introduction
2 Inelastic BGK Model
2.1 Kinetic Equations
2.2 Free Cooling (D = 0)
2.3 NESS (D = 0)
3 Basics of Inelastic Scattering Models
3.1 Boltzmann Equation as a Stochastic Process
3.2 Boltzmann Equation in Standard Form
3.3 Cooling and Driven Systems
3.4 Qualitative Analysis
3.5 Comments
4 Analysis of Inelastic Scattering Models
4.1 Homogeneous Cooling Laws
4.2 Scaling and Non-equilibrium Steady States
4.3 Comments
4.4 High Energy Tails
5 Inelastic Maxwell Models
5.1 Fourier Transform Method
5.2 Small-k Singularity of the Characteristic Function
5.3 Beyond Asymptotic Analysis
6 Conclusions and Perspectives
Chapter 2
1 Introduction
2 Setting Up the Problem
2.1 The Collision Model
2.2 The Boltzmann Equation
2.3 The Boltzmann Equation for the Homogeneous and Isotropic Case
3 Heuristic Analysis
3.1 Large Speeds
3.2 The Speed Ranges and Universality
4 The Near-Maxwellian Range of Speeds
5 Reduction of the Boltzmann Equation for the HCS
5.1 Derivation of the Reduced Equation
6 Concluding Remarks
Appendix A: The Gain Term for the HCS
A-1: Reduction of the Gain Term
A-2: The Gain Term for Large Speeds
A-2-1: Analysis of the {Gi ; i = 2}
A-
Appendix B: Remarks on the Mean Field Approach
Appendix C: Perturbation Theory
C-
C-2: Proof of the First Solubility Condition
C-3: The Asymptotic Ratio of the Gain to the Loss Term
Chapter 3
1 Introduction
2 Uniform Gases: One Dimension
2.1 The Freely Cooling Case
2.2 The Forced Case
3 Uniform Gases: Arbitrary Dimension
3.1 The Freely Cooling Case
3.2 The Forced Case
3.3 Velocity Correlations
4 Impurities
4.1 Model A
4.2 Model B
4.3 Velocity Autocorrelations
5 Mixtures
6 Lattice Gases
7 Conclusions
Chapter 4
1 Introduction
2 Instabilities of the Homogeneous Cooling State
2.1 The Homogeneous Cooling State
2.2 Instabilities of the Homogeneous Cooling State
3 A Starting Point: The Homogeneous Inelastic Maxwell Model
4 The One-Dimensional Gas
4.1 Molecular Dynamics
4.2 The Inelastic Lattice Maxwell Model
5 The Two-Dimensional Gas
5.1 Known Results from Molecular Dynamics
5.2 The Inelastic Maxwell Lattice Model
6 Conclusions
Chapter 5
1 Introduction
2 Isotropic Equation and Preliminary Result
3 A Recent Extension: Preliminaries
4 Complete Proof of the Conjecture
Chapter 6
1 Introduction
2 A Simple Example
3 Granular Gases of Viscoelastic Particles
4 Evaluation of Kinetic Integrals
4.1 De.nition of the Basic Integrals
4.2 Computation of the Basic Integrals
5 Computational Formula Manipulation to Evaluate Kinetic Integrals
6 Kinetic Integrals in the Kinetic Theory of Granular Gases
6.1 Homogeneous Cooling State
6.2 Inhomogeneous Granular Gases
7 Conclusion
Chapter 7
1 Introduction
2 Model
3 Kinetics
3.1 Scaling Behaviour
3.2 Kinetic Scenarios
4 Numerical Simulations
4.1 The Method
4.2 Numerical Results: Kinetics
4.3 Numerical Results: Distribution Functions
5 Discussion
Appendix: On the Collisional Average bc
Chapter 8
1 Introduction
2 One-Dimensional Waves
2.1 Hydrodynamic Model
2.2 Shock Wave
2.3 Expansion Wave
3 Two-Dimensional Waves
3.1 Flow around a Disk
3.2 Flow around a Wedge
4 Waves in Vibrated Granular Media
4.1 Basic Mechanisms of Vibro.uidization
4.2 Granular Gases Vibrated in Microgravity Field
4.3 Granular Gases Vibrated in Gravity Field
5 Summary
Chapter 9
1 Introduction
2 Nonlinear Boltzmann Equation and the Homogeneous Cooling State
3 Linearized Boltzmann Equation
4 Eigenvalue Problem
5 Navier-Stokes and Green-Kubo Expressions
5.1 Navier-Stokes Approximation
5.2 Green-Kubo Relations
6 Discussion
Chapter 10
1 Introduction
2 The Model Problem and Hydrodynamic Equations
3 Steady State Pro.les and Density Inversion
4 Formation of a Density Inversion: Low-Mach-Number Flow
5 Formation of a Density Inversion: Early Times
6 Discussion
Chapter 11
1 Introduction
2 Gas-Solids Interactions
3 Granular Transport Theory
4 Moment Method
5 Mixture Theory
6 Turbulence Modulation
7 Application
8 Comparisons between Predictions and Experiments
9 Conclusion
Chapter 12
1 Introduction
2 The Model
2.1 Dissipation on Collisions
2.2 Translational, Multiplicative Driving
2.3 Rotational Driving
3 Mean Field Evolution Equations
3.1 Smooth Particles – No Rotation
3.2 Rough Particles with Translational Driving
3.3 Rough Particles with Rotational Driving
4 Numerical Simulations
4.1 Approach to Steady State
4.2 Relaxation Time
4.3 Steady State Temperature
4.4 System Size Dependence
4.5 Steady State Clustering
5 Simulations with Rotation
6 Analytical Study of the Velocity Distribution


6.3 Rotations and Their Distribution
7 Summary and Conclusions
Chapter 13
Appendix: Density Fluctuations
Chapter 14
1 Introduction
2 Flux Model
3 More Than Two Compartments: Hysteresis
4 Coarsening and Sudden Death

6 Extensions and Applications
Chapter 15
1 Introduction
2 Description of the Simulations
2.1 The Variable Coe.cient of Restitution
2.2 The Other Simulation Parameters
3 Comparison of Simulation and Experiment
3.1 The Importance of the Variable Coe.cent of Restitution
3.2 The Importance of the Particle Number
4 E.ects of Clustering (n > 2 or 3)
4.1 Clustering without Gravity
4.2 Clustering with Gravity
5 Conclusions