粉体物理と惑星科学の接点ワークショップ

Weakly Nonlinear Analysis ofGranular Shear Flow
Kuniyasu Saitoh and Hisao Hayakawa
YITP, Kyoto University, Japan

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Contents
I.

Shear Band Formation of Sheared Granular Flow
I. Introduction - “What’s granular particle?” and “What’s granular flow?”
II. Granular Systems in Planetary Science
III. Kinetic Theory of Granular Gases
IV. Hydrodynamic equations of Granular Gases
V. Shear Band Formation ( DEM Simulation )VI. Shear Band Formation ( Analysis )VII. Analytic Solution for Steady StateVIII.Discussion and Conclusion

II. Linear Stability Analysis and Weakly Nonlinear Analysis of Sheared Granular Flow
I. Introduction
II. Linear Stability Analysis
III. Weakly Nonlinear Analysis
IV. Bifurcation Analysis
V. Numerical Analysis of the TDGL equation
VI. Discussion and Conclusion
2Ⅰ.
Shear Band Formation ofSheared Granular Flow3

Introduction – “What’s Granular Particle?”
Inelastic Collision
Restitution coefficient

0 < e <1

material constant?
Is that true?

Depends on Impact Speed, Situation, Size
Elasticity, Surface tension, viscoelasticity
Negative restitution coefficientis found in oblique impact ofnanoclusters
accepted to Phys. Rev. Lett. (2010)→ポスター発表

Negative!

e<04Introduction – “What’s Granular Flow?”Many-body system of Dissipative Particles
Always Non-equilibrium State
Coexistence of Static Region
Micro-polar Fluids ( Rotation )5

Granular Systems in Planetary Science
Proto-planetary Disk

Size Distribution in E-ringSelf-Gravity
Dilute System
Inelastic Collisions
Shear Flow

J. Schmidt, N. Brilliantov, F. Spahn, and S. Kempf,
Nature 451,451 685 (2008).
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Kinetic Theory for Granular Gases
Kinetic Theory

Boltzmann eq. for dissipative particles∂f 1 '' ''2+ u ⋅ ∇f = σ ∫ du1 ∫ dΩΘ(v r ⋅ n ) 2 f f1 − ff1 ∂teEnergy

not a collision invariant, energy loss rate

Velocity distribution function
Enskog method, Grad expansion methodf(0)Dependent on the system

Rotational degrees of freedom can be absorbed in e7Hydrodynamic equations of Granular Gases
Hydrodynamic Equations[Jenkins and Richman (1985),
Saitoh and Hayakawa (2007)]“Jenkins & Richman” without rotation
Area Fraction νVelocity Fields um2θ≡(c−u)
Granular Temperature2n(scaling units areshown in the next slide)8

Shear Band Formation
DEM Simulation

unNormal Force[Saitoh and Hayakawa (2007)]

Fn = −k nun − ηndun
dt

Tangential Forceutdu− kt ut − ηt t
Ft = dt − µ | Fn |If |Ft| < µ | Fn |If |Ft| ≥ µ | Fn |Setup
X-axis : Periodic boundaryY-axis : Bumpy boundary
Scaling Units
Mass : mLength : dTime : 2d / U9

Shear Band Formation
Results
Initial Condition
Configuration is at Random
Gaussian Velocity distribution
Transient Dynamics
Two narrow Shear Bands
Slowly moves to the center regionCollide and Merge
Steady State
A wide Shear Band

(Translation and Rotation)

Restitution coefficient : e = 0.85
Friction constant : μ = 0.2
The number of particles : N = 5000
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Shear Band Formation ( Analysis )YHydrodynamic Fields

average

Area Fraction ν (Y )Velocity Fields u = (u (Y ), w(Y ))Granular Temperature
2m nθ (Y ) ≡ ∑ (ci − u )2n i =1

Solve “Jenkins & Richman”
Boundary Conditions

energy loss rate11

Shear Band Formation ( Analysis )Results

Area Fraction

Initial Condition

Granular Temperature

Velocity field ( x direction )Kinetic Theory well reproduces the results,even if the density is high12Analytic Solution for Steady StateMean Area Fraction : 0.24d dν F()ν
 = G (ν )
dy 
dy 
F (ν ) ≡ p(ν )−3 / 22 κ (ν ) d (ν 2 g (ν )) 
1+(1+e)−()λν
2p()dνν τ  p(ν )(1 − e )ν g (ν )G (ν ) ≡ 2 −2π p(ν ) 3 / 2
 p  η (ν )
1/ 2

2

2

Pxy |boundary = τPyy |boundary = p

Transient Dynamics is Complex
Steady State is also well reproduced

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Discussion and Conclusion
Discussion

More Realistic Models
ex. Shear Flow of Cohesive Granular Particles
Influence of Self-Gravity
Quasi-2D system
Effects of Jamming and Rotation in the Dense Region
Fragmentation and Aggregation
Conclusion

Shear Band Formation is found in the DEM simulation
Analytic Solution of Steady Shear Band
Kinetic Theory works well, even if the Density is High14

Ⅱ.
Linear stability analysis andWeakly nonlinear analysis ofSheared Granular Flow15

Introduction

Linear Stability Analysis∧∧∧

Growth Rate

Re[σ ]Lφ = σ φWeakly Nonlinear Analysis
2Near the Neutral Mode σ = σ c + αε + L

φ − φ0 = φ ∝ eσt∂τ A = αA + γA + L
3σc = 0Pitchfork Bifurcation,
ex. Super-/Sub-critical bifurcation

σ c = iω

Hopf Bifurcation, ex. Limit Cycle
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Introduction

σc = 0α > 0, γ < 0A

αSupercritical
α > 0, γ > 0

SubcriticalA( A5 )αExternal field, ex. Gravity,サドルノード分岐
サドルノード分岐
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Introduction
Previous Works
Granular Gas P. Shukla and M. Alam(2009)

Homogeneous State is Unstable
Granular Gas:
Supercritical Bifurcation
Subcritical Bifurcation
Dense System:
Subcritical Bifurcation

Dense System E. Khain (2009)
hysteresis

18縮約によるによるアプローチ
縮約によるアプローチ（高密度な場合を除き）粉体集団のダイナミクスを記述する流体力学的な方程式

先行研究
P. Shukla and M. Alam (2009)

本研究
Newell & Whitehead(1969)の手続き

Reynolds & Potter(1967)の手続き※Y. Kuramoto, “Chemical Oscillations,Waves, and Turbulence”

Stuart-Landau型の振幅方程式

Ginzburg-Landau方程式問題点
問題点

特長特長

せん断方向の依存性を無視
せん断方向の依存性を無視

せん断方向の依存性を考慮
せん断方向の依存性を考慮

長さスケールが残らない
長さスケールが残らない

長さスケールを残し空間構造を捉える
長さスケールを残し空間構造を捉える
Lees-Edwards境界を適用
Lees-Edwards境界を適用

摂動に特別な境界条件を設定
摂動に特別な境界条件を設定

Setup and Basic equation

Setup
Lees-Edwards Boundary
Shear rate⋅

γ = 2d / L

Basic equation“Jenkins & Richman”
Hydrodynamic Field

Base State20

Setup and Basic equation

Convective Term

Stretched Coordinate

Base State21

Linear Stability Analysis

Linear equation
Symmetry

Fourier Transform

4×4 Real Matrix

22Linear Stability Analysis

Eigenvalue
Real part

Imaginary part23

Linear Stability Analysis

Neutral mode
Real number!

24

Weakly Nonlinear Analysis

Scaling
Hydrodynamic limiti.e.
Fixed Temperature

Expansion Parameter25

Weakly Nonlinear Analysis

Long Length Scale

Wave number

Long Time Scale … scaled by26

Weakly Nonlinear Analysis

Near the Neutral mode

α determines “Linearly Stable”
or “Linearly Unstable”
27

Weakly Nonlinear Analysis

Hydrodynamic equations

Disturbance Field

Neutral Solution
Disturbance Amplitude
28

Weakly Nonlinear Analysis

Zero eigenvector (Left)

Time Dependent Ginzburg-Landau equation

29Bifurcation Analysis
Coefficient ofAA20.53

Sub-critical
Super-critical

ε = 0.1

ν 0 < 0.53

Super-critical bifurcation

θ 0 = mU 2 / 4

ν 0 > 0.53

Sub-critical bifurcation30

Numerical Analysis of the TDGL equation

Super-critical
Bifurcation

31

Numerical Analysis of the TDGL equation

Sub-critical
Bifurcation
We assume− (α + γ ) A | A |432

Discussion and Conclusion
Discussion

Coefficient of A A

4Examine the Sub-critical Bifurcation.
In Experiment or the DEM simulation
Bifurcation Parameter U

Physical Boundary ConditionEffects of Jamming and Rotation in the Dense Region
Quasi-2D system

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Discussion and Conclusion
Conclusion

Derived the TDGL equation for
Sheared Granular Flowin the Hydrodynamic limit.
Sub-critical Bifurcation is foundin the Dense Region.
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Announcement
京都大学基礎物理学研究所主催 国際研究会

宇宙と物質の非平衡ダイナミクス
日程 ：2011年10月31日～11月3日

場所 ：京都大学 基礎物理学研究所
湯川記念館 Panasonicホール（予定）（粉体）ジェット ： Blandford氏, 平野氏, Nagel氏ほか
惑星リング ： Schmidt氏, Salo氏, Burns氏, Porcco氏ほか
MHD乱流 ： Balbus氏, 坪田氏, 後藤氏ほか
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