Kuniyasu Saitoh

Faculty of Engineering Technology,

University of Twente,

The Netherlands

*Instabilities in

Garnular Shear Flows

Contents I. Introduction

1. Instabilities in freely cooling state

2. Instabilities in granular shear flows

3. Aim & strategy

II. Molecular dynamics simulations

1. Setup & results

2. Kinetic theory of granular gases

III. Linear stability analysis

1. Basic equations & scaling units

2. Layering mode

3. Non-layering mode

IV. Weakly nonlinear analysis

1. 1-dimensional TDGL equation

2. Higher order amplitude equation

3. Bifurcation analysis

4. Hybrid approach & 2-dimensional TDGL equaiton

V. Discussion

1. Finite-size system

2. Hysteresis loop

VI. Conclusion

Instabilities in freely cooling state

Linear stability analysis

Hydrodynamic mode

kkkkk uuu| | , ,

Growth rate

)()()( kikk

Shear mode

Heat mode

Sound mode

0)( ),( kk

0)( ),( HH kk

0)( ,0)( SS kk

2

H 1 , ekk

2~ k

ku

| |

ku

ku

k

vortex

clustering

Instabilities in granular shear flows

Finite-size systems

Hydrodynamic limit

Linear stability analysis

Weakly nonlinear analysis

M. Alam & P. R. Nott, J. Fluid Mech. 377 (1998) 99

P. Shukla & M. Alam, Phys. Rev. Lett. 103 (2009) 068001

P. Shukla & M. Alam, J. Fluid Mech. 666 (2011) 204

Weakly nonlinear analysis (Ginzburg-Landau equation)

K. Saitoh & H. Hayakawa, Granular Matter 13 (2011) 697

Numerical solution of the Ginzburg-Landau equation

K. Saitoh & H. Hayakawa, AIP. Conf. Proc. 1501 (2012) 1001

K. Saitoh & H. Hayakawa, Phys. Fluids (2013) in press

Molecular dynamics simulation

K. Saitoh & H. Hayakawa, Phys. Rev. E 75 (2007) 021302

Strategy

Molecular dynamics

simulations

Observation

Kinetic theory

Linear & non-linear

analysis

Theory

Numerical

simulations of

amplitude eq.

Modeling

Comparison

Molecular dynamics simulations

Model

2-dimensional frictional granular particles

85.0e 2.0System size

5000N

Mean area fraction

ddLL 180180

12.00

Molecular dynamics simulations

Dense plug formation

Time development of the area fraction

(Saitoh & Hayakawa, 2007)

y

Bumpy wall

We also observed a similar plug formation of frictionless granular particles

under the Lees-Edwards boundary condition (Saitoh & Hayakawa, 2013)

a. Area fraction, velocity fields, and granular temperature .

b. Heat flux

c. Energy-sink term

d. Hydrostatic pressure & transport coefficients are the functions of .

Continuum equation

Equation of motion

Equation of energy

Granular hydrodynamic equations

Kinetic theory of granular gases

)( )( 1 2

2/3

1

2

jjue

Jenkins & Richman (1985)

Linear stability analysis

Granular hydrodynamic equations

Jenkins & Richman (1985)

2-dimensional frictionless disks

The Lees-Edwards boundary conditions

Scaling units

Mass

Length

Time

md

Udt 0

Shear rate 1

0

1

0 LdLU

: the ratio of particle’s diameter to gap

Hydrodynamic limit

1

Linear stability analysis

Homogeneous state

Finite temperature approximation

Hydrodynamic fields

)1(~0 O i.e. 221 e

),(),( 0 tt rr

T , , , wu

T000 ,0 , , y

T , , , wu

yxy

y

y

k

ti

t

kk

yik

k eAeAt rk

kr)(NL

)(

0

NLLL),(

Layering mode Non-layering mode

0 xk 0 xk

Kelvin mode xyx ktkkt , )( k

)1( ,0, , 22

00 ey uy : non-dimensionalized coordinate

Linear stability analysis

Linearized granular hydrodynamic equation

II )( tLdt

d

yxyxxyx kLtkkkLtkkkLtL 2

2

00 ,,)(

Layering mode

ykLtL ,0)( 0 : independent of time Matrix

Growth rate t

k ey

L

Eigenvalue problem

LL

0 ,0yy kkykL

0 xk

Perturbative calculations

0 LEigenvalue problem

Linear stability analysis

Wave vector qk y cf.) Clustering instabilities

, H

kk

2

2

10 MML

2

2

1

10

10~ ~ ~

Matrix

Eigenvalue

Right eigenvector

Left eigenvector

1*) We omit the superscript “L” and subscript “ky”.

2*) We have 4 eigenvalues and 4 eigenvectors.

Linear stability analysis

Dispersion relation

The eigenvalue with the maximum real part

4

4

2

221 ,0 qaqa

4

4

2

2

2)( qaqaq

42 2aaqc

The most unstable mode

2~ q

4~ q

Open circles

Numerical solution

Solid line

4

4

2

2)( qaqaq

Linear stability analysis

Eigenvectors (layering mode)

Open circles

Numerical solutions

Solid line

10)( q

1*) We also confirm good agreements of ~

Linear stability analysis

Eigenvectors (non-layering mode)

)(cos)()( 30

2

0

0NL

)( ttEJ

tEJ

pt

q

)(sin)()(1

1)(

)(13

21

2

NL

)( ttEt

tEt

tu t

q

)(sin)()(1

)()(1

13

21

2

NL

)( ttEt

ttE

tw t

q

)(cos)(2

)('

30

20NL

)( ttEJ

ptE

J

pt q

tetEtEtE2

~)( ),( ),( 321

NLNL )( tLdt

d

Weakly nonlinear analysis

Scaling of the eigenvalue and wave number

Long length & long time scales

2 qky

Note:

Long time scale

Long length scale

t 2

y

2

t

y

The most unstable solution

c.c. , LL

m c

c

iq

q eA

1*) Other modes can be suppressed.

t qyky

Weakly nonlinear analysis

Perturbative expansions

2

2

10 MML

2

2

1 AAA

Perturbative calculations

OO ,1 : absent

2O 0L

1 cqM 0~

1

L

1

L cc qq M

consistent

3O2L

1

L

13

L

1

2L

1

L

2

L

1

L || AANADAMAcc qq

2L

1

L

1

L

1

2L

12

L

1 || AAAdAA

3

LL

2

L

2

LLL ~ ,~ ,~ ,1~ NDdMcccccc qqqqqq

1D TDGL eq.

Weakly nonlinear analysis

Higher order calculations

4O

5O

3O2L

1

L

1

L

1

2L

12

L

1 || AAAdAA

L

2

2L

1

L

2

2L

1

L

2

2L

22

L

2 ||2 AAAAAdAA

4L

1

L

1

2L

2

L

1

L

3

2L

32

L

3 || AAAAAdAA

2

4LL2LLL2L

2

L |~

|~

|~

|~

~

~~

AAAAAdAA

L

3

2L

2

L

1

L ~

AAAA

Envelop function

Sum up

2

Weakly nonlinear analysis

Bifurcation analysis

Supercritical

Subcritical

|| A

super sub

0

Elastic & non-shear limit

i.e. Equilibrium gas 0

Weakly nonlinear analysis

Hybrid approach

22

2

2

12 || )( )( AAAdAdAdAA

2D TDGL eq.

New terms

cq )(q

)(q

c.c. ,, )(NL

)(

L

h zq

q

i

q eAc

Small deviation around the most unstable mode

)()( qqq c

The most unstable solution

c.c. )(L

m zq i

q eAc

c.c. )(NL

)(d zq

q

ieA

“Deviated” solution

Weakly nonlinear analysis

Time dependent diffusion coefficients

22

2

2

12 || )( )( AAAdAdAdAA

2D TDGL eq.

22

2 || AAAdAA 1D TDGL eq.

Zero

e~NL

)(q

Decay to zero

before 1)(1 d

)(2 d

Numerical solutions

Scheme

The 4th order Runge-Kutta method for the time-integration

The central difference method for the diffusion terms

Periodic boundary conditions in the sheared frame

Small perturbations with randomly chosen wave numbers

22

2

2

12 || )( )( AAAdAdAdAA

Numerical solutions

2D

TDGL

MD

(CG)

MD

Area fraction

Scaling function

Numerical solutions

Discussion

Finite-size systems

Hysteresis loop

0

|| A

sub

0

Does not exist! By definition

We couldn’t discuss hysteresis!

The ratio of a particle’s diameter to gap does not need to be small. Ld

2

2

01 e

The mean granular temperature

diverges in the elastic limit (e=1) Problem 1

is independent of and becomes another parameter.

What is a small parameter for the nonlinear analysis?

eProblem 2

The Fourier transformations could be questionable. Problem 3

Conclusion

Observation

Dense plug formations in 2-dimensional granular shear flows

are observed in both the bumpy & Lees-Edwards boundaries.

Theory

Granular hydrodynamic equations derived by the kinetic theory

can describe the dynamics of dense plug formations.

We perturbatively solved the linearized granular hydrodynamic

equations and found the hydrodynamic modes and eigenvalues.

From the weakly nonlinear analysis, we derived the 1D TDGL

equation and discussed the bifurcations of steady amplitude.

By taking the “hybrid” approach, we also introduced the 2D

TDGL equation with the time dependent diffusion coefficients.

Modeling

The 2D TDGL equation “qualitatively” describes the plug formations.